Physics
Physics B
Physics C: Mechanics
Physics C: Electricity and
MagnEtism
Course Description
Effective Fall 2012
AP Course Descriptions are updated regularly. Please visit AP Central ®
(apcentral.collegeboard.org) to determine whether a more recent Course
Description PDF is available.
The College Board
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AP Course Descriptions
AP Course Descriptions are updated regularly. Please visit AP Central®
(apcentral.collegeboard.org) to determine whether a more recent Course Description
PDF is available.
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Contents
About the AP Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Offering AP Courses and Enrolling Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How AP Courses and Exams Are Developed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How AP Exams Are Scored . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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AP Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
What We Are About: A Message from the Development Committee . . . . . . . . . . 4
The Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Course Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Instructional Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Laboratory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Importance and Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Implementation and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Documenting Laboratory Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Physics B Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Physics C Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Comparison of Topics in Physics B and Physics C . . . . . . . . . . . . . . . . . . . . . . . . 12
Content Outline for Physics B and Physics C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Learning Objectives for AP Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
The Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
The Free-Response Sections — Student Presentation . . . . . . . . . . . . . . . . . . . . . 39
Calculators and Equation Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Physics B Sample Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Answers to Physics B Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . . . . . 52
Physics B Sample Free-Response Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Physics C: Mechanics Sample Multiple-Choice Questions . . . . . . . . . . . . . . . . . . 62
Answers to Physics C: Mechanics Multiple-Choice Questions . . . . . . . . . . . . 66
Physics C: Mechanics Sample Free-Response Questions . . . . . . . . . . . . . . . . . . . 67
Physics C: Electricity and Magnetism Sample Multiple-Choice
Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Answers to Physics C: Electricity and Magnetism
Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Physics C: Electricity and Magnetism Sample Free-Response
Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Teacher Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
AP Central (apcentral.collegeboard.org) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Additional Resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
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About the AP® Program
AP ® enables students to pursue college-level studies while still in high school. Through
more than 30 courses, each culminating in a rigorous exam, AP provides willing and academically prepared students with the opportunity to earn college credit, advanced placement, or both. Taking AP courses also demonstrates to college admission officers that students have sought out the most rigorous course work available to them.
Each AP course is modeled upon a comparable college course, and college and university faculty play a vital role in ensuring that AP courses align with college-level standards. Talented and dedicated AP teachers help AP students in classrooms around the world develop and apply the content knowledge and skills they will need in college.
Each AP course concludes with a college-level assessment developed and scored by college and university faculty as well as experienced AP teachers. AP Exams are an essential part of the AP experience, enabling students to demonstrate their mastery of college-level course work. More than 90 percent of four-year colleges and universities in the United States grant students credit, placement, or both on the basis of successful AP Exam scores. Universities in more than 60 countries recognize AP Exam scores in the admission process and/or award credit and placement for qualifying scores. Visit www.collegeboard.org/ap/creditpolicy to view AP credit and placement policies at more than 1,000 colleges and universities.
Performing well on an AP Exam means more than just the successful completion of a course; it is a pathway to success in college. Research consistently shows that students who score a 3 or higher on AP Exams typically experience greater academic success in college and are more likely to graduate on time than otherwise comparable non-AP peers. Additional AP studies are available at www.collegeboard.org/
apresearchsummaries.
OfferingAPCoursesandEnrollingStudents
This course description details the essential information required to understand the objectives and expectations of an AP course. The AP Program unequivocally supports the principle that each school develops and implements its own curriculum that will enable students to develop the content knowledge and skills described here.
Schools wishing to offer AP courses must participate in the AP Course Audit, a process through which AP teachers’ syllabi are reviewed by college faculty. The AP Course Audit was created at the request of College Board members who sought a means for the College Board to provide teachers and administrators with clear guidelines on curricular and resource requirements for AP courses and to help colleges and universities validate courses marked “AP” on students’ transcripts. This process ensures that AP teachers’ syllabi meet or exceed the curricular and resource expectations that college and secondary school faculty have established for college-level courses. For more information on the AP Course Audit, visit www.collegeboard.org/apcourseaudit.
©
2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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HowAPCoursesandExamsAreDeveloped
AP courses and exams are designed by committees of college faculty and expert AP teachers who ensure that each AP subject reflects and assesses college-level expectations. AP Development Committees define the scope and expectations of the course, articulating through a curriculum framework what students should know and be able to do upon completion of the AP course. Their work is informed by data collected from a range of colleges and universities to ensure that AP coursework reflects current scholarship and advances in the discipline. To find a list of each subject’s current AP Development Committee members, please visit apcentral.collegeboard.org/developmentcommittees. The AP Development Committees are also responsible for drawing clear and wellarticulated connections between the AP course and AP Exam — work that includes designing and approving exam specifications and exam questions. The AP Exam development process is a multi-year endeavor; all AP Exams undergo extensive review, revision, piloting, and analysis to ensure that questions are high quality and fair, and that there is an appropriate spread of difficulty across the questions.
Throughout AP course and exam development, the College Board gathers feedback from various stakeholders in both secondary schools and higher education institutions. This feedback is carefully considered to ensure that AP courses and exams are able to provide students with a college-level learning experience and the opportunity to demonstrate their qualifications for advanced placement upon college entrance.
HowAPExamsAreScored
The exam scoring process, like the course and exam development process, relies on the expertise of both AP teachers and college faculty. While multiple-choice questions are scored by machine, the free-response questions are scored by thousands of college faculty and expert AP teachers at the annual AP Reading. AP Exam Readers are thoroughly trained, and their work is monitored throughout the Reading for fairness and consistency. In each subject, a highly respected college faculty member fills the role of Chief Reader, who, with the help of AP Readers in leadership positions, maintains the accuracy of the scoring standards. Scores on the free-response questions are weighted and combined with the weighted results of the computer-scored multiplechoice questions. These composite, weighted raw scores are converted into the reported AP Exam scores of 5, 4, 3, 2, and 1.
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The score-setting process is both precise and labor intensive, involving numerous psychometric analyses of the results of a specific AP Exam in a specific year and of the particular group of students who took that exam. Additionally, to ensure alignment with college-level standards, part of the score-setting process involves comparing the performance of AP students with the performance of students enrolled in comparable courses in colleges throughout the United States. In general, the AP composite score points are set so that the lowest raw score needed to earn an AP Exam score of 5 is equivalent to the average score among college students earning grades of A in the college course. Similarly, AP Exam scores of 4 are equivalent to college grades of A–, B+, and B. AP Exam scores of 3 are equivalent to college grades of B–, C+, and C.
APScore
5
4
3
2
1
Qualification
Extremely well qualified
Well qualified
Qualified
Possibly qualified
No recommendation
AdditionalResources
Visit apcentral.collegeboard.org for more information about the AP Program.
©
2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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AP Physics
Introduction
What We Are About: A Message from the Development
Committee
The AP Physics Development Committee recognizes that curriculum, course content,
and assessment of scholastic achievement play complementary roles in shaping
education at all levels. The committee believes that assessment should support and
encourage the following broad instructional goals:
1. Physics knowledge — Basic knowledge of the discipline of physics, including
phenomenology, theories and techniques, concepts and general principles
2. Problem solving — Ability to ask physical questions and to obtain solutions to
physical questions by use of qualitative and quantitative reasoning and by
experimental investigation
3. Student attributes — Fostering of important student attributes, including
appreciation of the physical world and the discipline of physics, curiosity,
creativity, and reasoned skepticism
4. Connections — Understanding connections of physics to other disciplines and to
societal issues
The first three of these goals are appropriate for the AP and introductory-level college
physics courses that should, in addition, provide a background for the attainment of
the fourth goal.
The AP Physics Exams have always emphasized achievement of the first two goals.
Over the years, the definitions of basic knowledge of the discipline and problem
solving have evolved. The AP Physics courses have reflected changes in college
courses, consistent with our primary charge. We have increased our emphasis on
physical intuition, experimental investigation, and creativity. We include more openended questions in order to assess students’ ability to explain their understanding of
physical concepts. We ­structure questions that stress the use of mathematics to
illuminate the physical situation rather than to show manipulative abilities.
The committee is dedicated to developing exams that can be graded fairly and
consistently and that are free of ethnic, gender, economic, or other bias. We operate
under practical constraints of testing methods, allotted time and large numbers of
students at widely spread geographical locations. In spite of these constraints, the
committee strives to design exams that promote excellent and appropriate instruction
in physics.
The Courses
The AP Physics Exams are designed to test student achievement in the AP Physics
courses described in this book. These courses are intended to be representative of
courses commonly offered in colleges and universities, but they do not necessarily
correspond precisely to courses at any particular institution. The aim of an AP
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secondary school course in physics should be to develop the students’ abilities to do
the following:
1. Read, understand, and interpret physical information — verbal, ­mathematical,
and graphical
2. Describe and explain the sequence of steps in the analysis of a particular
physical phenomenon or problem; that is,
a. describe the idealized model to be used in the analysis, including simplifying
assumptions where necessary;
b. state the concepts or definitions that are applicable;
c. specify relevant limitations on applications of these principles;
d. carry out and describe the steps of the analysis, verbally, or ­mathematically;
and
e. interpret the results or conclusions, including discussion of ­particular cases
of special interest
3. Use basic mathematical reasoning — arithmetic, algebraic, geometric, trigonometric, or calculus, where appropriate — in a physical situation or problem
4. Perform experiments and interpret the results of observations, including making
an assessment of experimental uncertainties
In the achievement of these goals, concentration on basic principles of physics and
their applications through careful and selective treatment of well-chosen areas is more
important than superficial and encyclopedic coverage of many detailed topics. Within
the general framework outlined on pages 13–15, teachers may exercise some freedom in
the choice of topics.
In the AP Physics Exams, an attempt is made through the use of ­multiple-choice
and free-response questions to determine how well these goals have been achieved by
the student either in a conventional course or through independent study. The level of
the student’s achievement is assigned an AP Exam score of 1 to 5, and many colleges
use this score alone as the basis for placement and credit decisions.
Introductory college physics courses typically fall into one of three categories,
designated as A, B, and C in the following discussion.
Category A includes courses in which major concepts of physics are covered
without as much mathematical rigor as in more formal courses, such as Category B
and Category C, which are described below. The emphasis in Category A courses is on
developing a qualitative conceptual understanding of general principles and models
and on the nature of scientific inquiry. Some courses may also view physics primarily
from a cultural or historical perspective. Category A courses are generally intended for
­students not majoring in a science-related field. The level of mathematical sophistication
usually includes some algebra and may extend to simple trigonometry, but rarely
beyond. These courses vary widely in content and approach, and at present there is
no AP course or exam in this category. A high school version of a Category A course
that concentrates on conceptual development and that provides an enriching laboratory
experience may be taken by students in the ninth or tenth grade and should provide
the first course in physics that prepares them for a more mathematically ­rigorous AP
Physics B or C course.
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Category B courses build on the conceptual understanding attained in a first course
in physics, such as the Category A course described previously. These courses provide
a systematic development of the main principles of physics, emphasizing problem
solving and helping students develop a deep understanding of physics concepts. It is
assumed that students are familiar with algebra and trigonometry, although some
theoretical developments may use basic concepts of calculus. In most colleges, this is
a one-year ­terminal course including a laboratory component and is not the usual
preparation for more advanced physics and engineering courses. However, Category B
courses often provide a foundation in physics for students in the life sciences, premedicine, and some applied sciences, as well as other fields not directly related to science.
AP Physics B is intended to be equivalent to such courses.
Category C courses also build on the conceptual understanding attained in a first
course in physics, such as the Category A course described above. These courses
normally form the college sequence that serves as the ­foundation in physics for
students majoring in the physical sciences or engineering. The sequence is parallel to
or preceded by mathematics courses that include calculus. Methods of calculus are
used in formulating physical principles and in applying them to physical problems.
The sequence is more intensive and analytic than in Category B courses. Strong
emphasis is placed on solving a variety of challenging problems, some requiring
calculus, as well as continuing to develop a deep understanding of physics concepts. A
Category C sequence may be a very ­intensive one-year course in college but often will
extend over one and one-half to two years, and a laboratory component is also
included. AP Physics C is intended to be equivalent to part of a Category C sequence
and covers two major areas: mechanics, and electricity and magnetism, with equal
emphasis on both.
In certain colleges and universities, other types of unusually high-level introductory
courses are taken by a few selected students. Selection of students for these courses is
often based on results of AP Exams, other college admission information, or a collegeadministered exam. The AP Exams are not designed to grant credit or exemption for
such high-level courses but may facilitate admission to them.
Course Selection
It is important for those teaching and advising AP students to consider the relation of
AP courses to a student’s college plans. In some circumstances it is advantageous to
take the AP Physics B course. The student may be interested in studying physics as a
basis for more advanced work in the life sciences, medicine, geology, and related
areas, or as a component in a nonscience college program that has science requirements. Credit or advanced placement for the Physics B course provides the student
with an opportunity either to have an accelerated college program or to meet a basic
science requirement; in either case the student’s college program may be enriched.
Access to an intensive physics sequence for physics or science majors is another
opportunity that may be available.
For students planning to specialize in a physical science or in engineering, most
colleges require an introductory physics sequence that includes courses equivalent to
Physics C. Since a previous or concurrent course in calculus is often required of
students taking Physics C, students who expect advanced placement or credit for
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either Physics C exam should attempt an AP course in calculus as well; otherwise,
placement in the next-in-sequence physics course may be delayed or even denied.
Either of the AP Calculus courses, Calculus AB or Calculus BC, should provide an
acceptable basis for students preparing to major in the physical sciences or engineering, but Calculus BC is recommended. Therefore, if such students must choose
between AP Physics or AP Calculus while in high school, they should probably choose
AP Calculus.
There are three separate AP Physics Exams, Physics B, Physics C: Mechanics and
Physics C: Electricity and Magnetism. Each exam contains multiple-choice and freeresponse questions. The Physics B Exam is for students who have taken a Physics B
course or who have mastered the material of this course through independent study.
The Physics B Exam covers topics in mechanics, electricity and magnetism, fluid
mechanics and thermal physics, waves and optics, and atomic and nuclear physics; a
single exam score is reported. Similarly, the two Physics C Exams correspond to the
Physics C course sequence. One exam covers mechanics; the other covers electricity
and magnetism. Students may take either or both exams, and separate scores are
reported.
Further descriptions of the AP Physics courses and their corresponding exams in
terms of topics, level, mathematical rigor, and ­typical textbooks are presented in the
pages that follow. Information about organizing and conducting AP Physics courses,
of interest to both beginning and experienced AP teachers, may be found on the AP
Physics home pages on AP Central (apcentral.collegeboard.org). These pages include
practical advice from successful AP teachers. The 2009 AP Physics B and Physics C
Released Exams book contains the complete exams, with solutions and grading
standards for the free-response sections and sample student responses, as well as
statistical data on student ­performance. For information about ordering these
publications and others, see page 81.
Instructional Approaches
It is strongly recommended that both Physics B and Physics C be taught as
second-year physics courses. A first-year physics course aimed at developing a
thorough understanding of important physical ­principles and that permits students to
explore concepts in the laboratory provides a richer experience in the process of
science and better prepares them for the more analytical approaches taken in AP
courses.
However, secondary school programs for the achievement of AP course goals can
take other forms as well, and the imaginative teacher can design approaches that best
fit the needs of his or her students. In some schools, AP Physics has been taught
successfully as a very intensive first-year course; but in this case there may not be
enough time to cover the material in sufficient depth to reinforce the students’
conceptual understanding or to provide adequate laboratory experiences. This
approach can work for highly motivated, able students but is not generally recommended. Independent study or other first-year physics courses supplemented with
extra work for individual motivated students are also possibilities that have been
successfully implemented.
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If AP Physics is taught as a second-year course, it is recommended that the course
meet for at least 250 minutes per week (the equivalent of a 50-minute period every
day). However, if it is to be taught as a first-year course, approximately 90 minutes per
day (450 minutes per week) is ­recommended in order to devote sufficient time to
study the material to an appropriate depth and allow time for labs.
In a school that uses block scheduling, it is strongly recommended that AP Physics B
be scheduled to extend over an entire year. A one-year AP course should not be taught
in one semester, as this length of time is insufficient for students to properly assimilate
and understand the important concepts of physics that are covered in the syllabus.
Each of the Physics C courses, but not both, can be taught in one semester.
Whichever approach is taken, the nature of the AP course requires teachers to
spend time on the extra preparation needed for both class and laboratory. AP teachers
should have a teaching load that is adjusted accordingly.
Laboratory
Importance and Rationale
Laborator y experience must be part of the education of AP Physics students
and should be included in all AP Physics courses, just as it is in introductory
college physics courses. In textbooks and problems, most attention is paid to
idealized situations: friction is often assumed to be constant or absent; meters read
true values; heat insulators are perfect; gases follow the ideal gas equation. It is in the
laboratory that the validity of these assumptions can be questioned, because there the
student meets nature as it is rather than in idealized form. Consequently, AP students
should be able to:
• design experiments;
• observe and measure real phenomena;
• organize, display, and critically analyze data;
• analyze sources of error and determine uncertainties in measurement;
• draw inferences from observations and data; and
• communicate results, including suggested ways to improve experiments and
proposed questions for further study.
Laboratory experience is also important in helping students understand the topics
being considered. Thus it is valuable to ask students to write informally about what
they have done, observed, and concluded, as well as for them to keep well-organized
laboratory notebooks.
Students need to be proficient in problem solving and in the application of
fundamental principles to a wide variety of situations. Problem-solving ability can be
fostered by investigations that are somewhat nonspecific. Such investigations are often
more interesting and valuable than “cookbook” experiments that merely investigate a
well-established relationship and can take important time away from the rest of the
course.
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© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Some questions or parts of questions on each AP Physics Exam deal with labrelated skills, such as design of experiments, data analysis, and error analysis, and
may distinguish between students who have had laboratory experience and those who
have not. In addition, understanding gained in the laboratory may improve students’
test performance overall.
Implementation and Recommendations
Laboratory programs in both college courses and AP courses differ widely, and there
is no clear evidence that any one approach is necessarily best. This diversity of
approaches should be encouraging to the high school teacher of an AP course. The
success of a given program depends strongly on the interests and enthusiasm of the
teacher and on the general ability and motivation of the students involved.
Although programs differ, the AP Physics Development Committee has made some
recommendations in regard to school resources and scheduling. Since an AP
course is a college course, the equipment and time allotted to laboratories
should be similar to that in a college course. Therefore, school administrators
should realize the implications, in both cost and time, of incorporating
serious laboratories into their program. Schools must ensure that students
have access to scientific equipment and all materials necessar y to conduct
hands-on, college-level physics laborator y investigations as outlined in the
teacher’s course syllabus.
In addition to equipment commonly included in college labs, students in AP Physics
should have adequate and timely access to computers that are connected to the Internet
and its many online resources. Students should also have access to computers with
appropriate sensing devices and software for use in gathering, graphing, and analyzing
laboratory data and writing reports. Although using computers in this way is a useful
activity and is encouraged, some initial experience with gathering, graphing, and
manipulating data by hand is also important so that students attain a better feel for the
physical realities involved in the experiments. And it should be emphasized that simulating an experiment on a computer cannot adequately replace the actual, hands-on
experience of doing an experiment.
Flexible or modular scheduling is best in order to meet the time requirements
identified in the course outline. Some schools are able to assign daily double periods
so that laboratory and quantitative problem-solving skills may be fully developed.
A weekly extended or double laboratory period is recommended for labs. It is not
advisable to attempt to complete high-quality AP laboratory work entirely within
standard 45- to 50-minute periods.
If AP Physics is taught as a second-year physics course, the AP labs should build on
and extend the lab experiences of the first-year course. The important criterion is that
students completing an AP Physics course must have had laboratory experiences that
are roughly equivalent to those in a comparable introductory college course.
Past surveys of introductory college physics courses, both noncalculus and
calculus-based, have revealed that on average about 20 percent of the total course
credit awarded can be attributed to lab performance; from two to three hours per
week are typically devoted to laboratory activities. Secondary schools may have
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difficulty scheduling this much weekly time for lab. However, the college academic
year typically contains fewer weeks than the secondary school year, so AP teachers
may be able to schedule a few more lab periods during the year than can colleges.
Also, college faculty have reported that some lab time occasionally may be used for
other purposes. Nevertheless, in order for AP students to have sufficient time for lab,
at least one double or extended period per week is recommended for all AP Physics
courses.
Laboratory activities in colleges and AP courses can involve different levels of
student involvement. They can generally be classified as: (1) prescribed or “cookbook,”
(2) limited investigations with some direction provided and (3) open investigations
with little or no direction provided. While many college professors believe that labs in
the latter two categories have more value to students, they report often being limited
in their ability to institute them by large class sizes and other factors. In this respect,
AP teachers often have an advantage in being able to offer more open-ended labs to
their students.
In past surveys, colleges have cited use of the following techniques to assess
student lab performance: lab reports, direct observation, written tests designed
specifically for lab, lab-related questions on regular lecture tests, lab practical exams,
and maintenance of lab notebooks. When the colleges assessed laboratory skills with
written test questions, they reported attempting to assess the following skills in order
of decreasing frequency: analysis of data, analysis of errors, design of experiments,
and evaluation of experiments and suggestions for future investigations.
A more detailed laboratory guide is available and can be ordered through AP Central.
This guide contains descriptions of a number of experiments that typify the type and
level of skills that should be developed by AP students in conducting laboratory
investigations. The experiments are not mandatory; they can be modified or similar
experiments substituted as long as they assist the student in developing these skills.
Additional suggestions for the laboratory can be found on the AP Physics course
home pages on AP Central (apcentral.collegeboard.org).
Documenting Laboratory Experience
The laboratory is important for both AP and college students. Students who have had
laboratory experience in high school will be in a better position to validate their AP
courses as equivalent to the corresponding college courses and to undertake the
laboratory work in more advanced courses with greater confidence. Most college
placement policies assume that students have had laboratory experience, and students
should be prepared to show evidence of their laboratory work in case the college asks
for it. Such experience should be documented for the AP course by keeping a lab
notebook or a portfolio of lab reports. Students should be encouraged to keep copies
of this work and any other work from previous lab experience. Presenting evidence of
adequate college-level laboratory experience to the colleges they attend, as an adjunct
to their AP scores, can be very useful to students if they desire credit for or exemption
from an introductory college course that includes a laboratory. Although colleges can
expect that most entering AP students have been exposed to many of the same
laboratory experiments performed by their own introductory students, individual
consultation with students is often used to help determine the nature of their
laboratory experience.
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© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Physics B Course
The Physics B course includes topics in both classical and modern physics. A
knowledge of algebra and basic trigonometry is required for the course; the basic
ideas of calculus may be introduced in connection with physical concepts, such as
acceleration and work. Understanding of the basic principles involved and the ability
to apply these principles in the solution of problems should be the major goals of the
course. Consequently, the course should utilize guided inquiry and student-centered
learning to foster the development of critical thinking skills.
Physics B should provide instruction in each of the following five content areas:
Newtonian mechanics, fluid mechanics and thermal physics, electricity and magnetism,
waves and optics, and atomic and nuclear physics. A content outline and percentage
goals for covering each major topic in the exam are on pages 13–15. A more detailed
topic outline is contained in the “Learning Objectives for AP Physics,” which starts on
page 17.
Many colleges and universities include additional topics in their survey courses.
Some AP teachers may wish to add supplementary material to a Physics B course.
Many teachers have found that a good time to do this is late in the year, after the
AP Exams have been given.
The Physics B course should also include a hands-on laboratory component
comparable to introductory college-level physics laboratories, with a minimum of 12
student-conducted laboratory investigations representing a variety of topics covered in
the course. Each student should complete a lab notebook or portfolio of lab reports.
The school should ensure that each student has a copy of a college-level textbook
(supplemented when necessary to meet the curricular requirements) for individual
use inside and outside of the classroom. A link to a list of examples of acceptable
textbooks can be found on the Physics B course home page on the AP Central Web
site.
Physics C Courses
There are two AP Physics C courses — Physics C: Mechanics and Physics C: Electricity
and Magnetism, each corresponding to approximately a semester of college work.
Mechanics is typically taught first, and some AP teachers may choose to teach this
course only. If both courses are taught over the course of a year, approximately equal
time should be given to each. Both courses should utilize guided inquiry and studentcentered learning to foster the development of critical thinking skills and should use
introductory differential and integral calculus throughout the course.
Physics C: Mechanics should provide instruction in each of the following six
content areas: kinematics; Newton’s laws of motion; work, energy and power; systems
of particles and linear momentum; circular motion and rotation; and oscillations and
gravitation.
Physics C: Electricity and Magnetism should provide instruction in each of the
following five content areas: electrostatics; conductors, capacitors and dielectrics;
electric circuits; magnetic fields; and electromagnetism.
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Content outlines for both courses and percentage goals for covering each major
topic in the exams are on pages 13–15. A more detailed topic outline is contained in
the “Learning Objectives for AP Physics,” which start on page 17.
Most colleges and universities include in similar courses additional topics such as
wave motion, kinetic theory and thermodynamics, optics, alternating current circuits,
or special relativity. Although wave motion, optics and kinetic theory and thermodynamics are usually the most commonly included, there is little uniformity among
such offerings, and these topics are not included in the Physics C Exams. The
Development Committee recommends that supplementary material be added to
Physics C when it is possible to do so. Many teachers have found that a good time to
do this is late in the year, after the AP Exams have been given.
Each Physics C course should also include a hands-on laboratory component
comparable to a semester-long introductory college-level physics laboratory. Students
should spend a minimum of 20 percent of instructional time engaged in hands-on
laboratory work. Each student should complete a lab notebook or portfolio of lab
reports.
The school should ensure that each student has a calculus-based college-level
textbook (supplemented when necessary to meet the curricular requirements) for
individual use inside and outside of the classroom. A link to lists of examples of
acceptable textbooks can be found on the Physics C course home pages on the
AP Central website.
Comparison of Topics in Physics B and Physics C
To serve as an aid for devising AP Physics courses and to more clearly identify the
specifics of the exams, a detailed topical structure has been developed that relies
heavily on information obtained in college surveys. The general areas of physics are
subdivided into major categories on pages 13–15, and for each category the percentage
goals for each exam are given. These goals should serve only as a guide and should
not be construed as reflecting the proportion of course time that should be devoted to
each category.
Also, for each major category, some important subtopics are listed. The checkmarks
indicate the subtopics that may be covered in each exam. Questions for the exam will
come from these subtopics, but not all of the subtopics will necessarily be included in
every exam, just as they are not necessarily included in every AP or college course.
It should be noted that although fewer topics are covered in Physics C than in
Physics B, they are covered in greater depth and with greater analytical and mathematical sophistication, including calculus applications.
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© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Content Outline for Physics B and Physics C
A more detailed topic outline is contained in the “Learning Objectives for
AP Physics,” which follow this outline.
Percentage Goals for Exams
Physics B
Physics C:
Content Area
Mechanics
I. Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35%
A. Kinematics (including vectors, vector algebra, components of vectors, coordinate systems,
displacement, velocity, and acceleration)
1. Motion in one dimension
2. Motion in two dimensions, including
projectile motion
7%
100%
18%
√
√
√
√
B. Newton’s laws of motion
1. Static equilibrium (first law)
2. Dynamics of a single particle (second law)
3. Systems of two or more objects (third law) 9%
√
√
√
20%
√
√
√
C. Work, energy, power
1. Work and work–energy theorem
2. Forces and potential energy
3. Conservation of energy
4. Power
5%
√
√
√
√
14%
√
√
√
√
D. Systems of particles, linear momentum
4%
1. Center of mass
2. Impulse and momentum
√
3. Conservation of linear momentum, √
collisions
12%
√
√
√
E. Circular motion and rotation
4%
1. Uniform circular motion
√
2. Torque and rotational statics
√
3. Rotational kinematics and dynamics
4. Angular momentum and its conservation
18%
√
√
√
√
F. Oscillations and gravitation
6%
1. Simple harmonic motion (dynamics and √
energy relationships)
2. Mass on a spring
√
3. Pendulum and other oscillations
√
4. Newton’s law of gravity
√
5. Orbits of planets and satellites
a. Circular
√
b. General
18%
√
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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√
√
√
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Content Area
Percentage Goals for Exams
Physics B
II. Fluid Mechanics and Thermal Physics. . . . . . . . . . . . . . . . . . 15%
A. Fluid Mechanics
1. Hydrostatic pressure
2. Buoyancy
3. Fluid flow continuity
4. Bernoulli’s equation
6%
√
√
√
√
B. Temperature and heat
1. Mechanical equivalent of heat
2. Heat transfer and thermal expansion
2%
√
√
C. Kinetic theory and thermodynamics
7%
1. Ideal gases
a. Kinetic model
√
b. Ideal gas law
√
2. Laws of thermodynamics
a. First law (including processes on √
pV diagrams)
b. Second law (including heat engines) √
Physics C:
Electricity and
Magnetism
III. Electricity and Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 25%
100%
A. Electrostatics
5%
1. Charge and Coulomb’s law
√
2. Electric field and electric potential (including
√
point charges)
3. Gauss’s law
4. Fields and potentials of other charge distributions
30%
√
√
B. Conductors, capacitors, dielectrics
4%
1. Electrostatics with conductors
√
2. Capacitors
a. Capacitance
√
b. Parallel plate
√
c. Spherical and cylindrical
3. Dielectrics
14%
√
C. Electric circuits
7%
1. Current, resistance, power
√
2. Steady-state direct current circuits with
√
batteries and resistors only
3. Capacitors in circuits
a. Steady state
√
b. Transients in rc circuits
20%
√
√
14
√
√
√
√
√
√
√
√
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Percentage Goals for Exams
Physics B
Physics C:
Electricity and
Content Area
Magnetism
D. Magnetic Fields
4%
1. Forces on moving charges in magnetic fields
√
2. Forces on current-carrying wires in
√
magnetic fields
3. Fields of long current-carrying wires
√
4. Biot–Savart law and Ampere’s law
20%
√
√
E. Electromagnetism
5%
1. Electromagnetic induction (including
√
Faraday’s law and Lenz’s law)
2. Inductance (including lr and lc circuits)
3. Maxwell’s equations
16%
√
√
√
√
√
IV. Waves and Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15%
A. Wave motion (including sound)
1. Traveling waves
2. Wave propagation
3. Standing waves
4. Superposition
5%
√
√
√
√
B. Physical optics
1. Interference and diffraction
2. Dispersion of light and the electromagnetic
spectrum
5%
√
√
C. Geometric optics
1. Reflection and refraction
2. Mirrors
3. Lenses
5%
√
√
√
V. Atomic and Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 10%
A. Atomic physics and quantum effects
1. Photons, the photoelectric effect,
Compton scattering, x-rays
2. Atomic energy levels
3. Wave-particle duality
7%
√
B. Nuclear physics
1. Nuclear reactions (including conservation of mass number and charge)
2. Mass–energy equivalence
3%
√
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
√
√
√
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Laboratory and experimental situations: Each exam will include one or more questions
or parts of questions posed in a laboratory or experimental setting. These questions are
classified according to the ­content area that provides the setting for the situation, and
each content area may include such questions. These questions generally assess some
understanding of content as well as experimental skills, as described on the following
pages.
Miscellaneous: Each exam may include occasional questions that overlap several
major topical areas or questions on miscellaneous topics such as identification of
vectors and scalars, vector mathematics, graphs of functions, history of physics, or
contemporary topics in physics.
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© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
Learning Objectives for AP Physics
These course objectives are intended to elaborate on the content outline for Physics B and Physics C. In
addition to the five major content areas of physics, objectives are included now for laboratory skills, which
have become an important part of the AP Physics Exams.
The objectives listed below are generally representative of the cumulative content of recently
administered exams, although no single exam can cover them all. The checkmarks indicate the objectives
that may be covered in either the Physics B or Physics C Exams.
It is reasonable to expect that future exams will continue to sample primarily from among these
objectives. However, there may be an occasional question that is within the scope of the included topics but
is not specifically covered by one of the listed objectives. Questions may also be based on variations or
combinations of these objectives, rephrasing them but still assessing the essential concepts.
The objectives listed below are continually revised to keep them as current as possible with the content
outline and the coverage of the exams.
Objectives for the AP® Physics Courses
I. NEWTONIAN MECHANICS
A. Kinematics (including vectors, vector algebra, components of vectors, coordinate
systems, displacement, velocity, and acceleration)
1. Motion in one dimension
a) Students should understand the general relationships among position, velocity, and
acceleration for the motion of a particle along a straight line, so that:
(1) Given a graph of one of the kinematic quantities, position, velocity, or
acceleration, as a function of time, they can recognize in what time intervals the
other two are positive, negative, or zero and can identify or sketch a graph of
each as a function of time.
(2) Given an expression for one of the kinematic quantities, position, velocity or
acceleration, as a function of time, they can determine the other two as a
function of time, and find when these quantities are zero or achieve their
maximum and minimum values.
b) Students should understand the special case of motion with constant acceleration,
so they can:
(1) Write down expressions for velocity and position as functions of time, and
identify or sketch graphs of these quantities.
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(2) Use the equations u = u0 + at , x = x0 + u0 t + at 2 , and
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u 2 = u0 2 + 2a ( x - x0 ) to solve problems involving one-dimensional motion
with constant acceleration.
c) Students should know how to deal with situations in which acceleration is a
specified function of velocity and time so they can write an appropriate differential
equation and solve it for u t by separation of variables, incorporating correctly a
given initial value of u .
AP Course
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© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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Objectives for the AP® Physics Courses
AP Course
B
C
2. Motion in two dimensions, including projectile motion
a) Students should be able to add, subtract, and resolve displacement and velocity
vectors, so they can:
(1) Determine components of a vector along two specified, mutually perpendicular
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axes.
(2) Determine the net displacement of a particle or the location of a particle relative
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to another.
(3) Determine the change in velocity of a particle or the velocity of one particle
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relative to another.
b) Students should understand the general motion of a particle in two dimensions so
that, given functions x(t) and y(t) which describe this motion, they can determine
the components, magnitude, and direction of the particle’s velocity and acceleration
as functions of time.
c) Students should understand the motion of projectiles in a uniform gravitational
field, so they can:
(1) Write down expressions for the horizontal and vertical components of velocity
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and position as functions of time, and sketch or identify graphs of these
components.
(2) Use these expressions in analyzing the motion of a projectile that is projected
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with an arbitrary initial velocity.
B. Newton’s laws of motion
1. Static equilibrium (first law)
Students should be able to analyze situations in which a particle remains at rest, or
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moves with constant velocity, under the influence of several forces.
2. Dynamics of a single particle (second law)
a) Students should understand the relation between the force that acts on an object
and the resulting change in the object’s velocity, so they can:
(1) Calculate, for an object moving in one dimension, the velocity change that
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results when a constant force F acts over a specified time interval.
(2) Calculate, for an object moving in one dimension, the velocity change that
results when a force F(t) acts over a specified time interval.
(3) Determine, for an object moving in a plane whose velocity vector undergoes a
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specified change over a specified time interval, the average force that acted on
the object.
b) Students should understand how Newton’s Second Law, Â F = Fnet = ma ,
applies to an object subject to forces such as gravity, the pull of strings, or contact
forces, so they can:
(1) Draw a well-labeled, free-body diagram showing all real forces that act on the
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object.
(2) Write down the vector equation that results from applying Newton’s Second
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Law to the object, and take components of this equation along appropriate axes.
c) Students should be able to analyze situations in which an object moves with
specified acceleration under the influence of one or more forces so they can
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determine the magnitude and direction of the net force, or of one of the forces that
makes up the net force, such as motion up or down with constant acceleration.
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Objectives for the AP® Physics Courses
d) Students should understand the significance of the coefficient of friction, so they
can:
(1) Write down the relationship between the normal and frictional forces on a
surface.
(2) Analyze situations in which an object moves along a rough inclined plane or
horizontal surface.
(3) Analyze under what circumstances an object will start to slip, or to calculate the
magnitude of the force of static friction.
e) Students should understand the effect of drag forces on the motion of an object, so
they can:
(1) Find the terminal velocity of an object moving vertically under the influence of
a retarding force dependent on velocity.
(2) Describe qualitatively, with the aid of graphs, the acceleration, velocity, and
displacement of such a particle when it is released from rest or is projected
vertically with specified initial velocity.
(3) Use Newton’s Second Law to write a differential equation for the velocity of
the object as a function of time.
(4) Use the method of separation of variables to derive the equation for the velocity
as a function of time from the differential equation that follows from Newton’s
Second Law.
(5) Derive an expression for the acceleration as a function of time for an object
falling under the influence of drag forces.
3. Systems of two or more objects (third law)
a) Students should understand Newton’s Third Law so that, for a given system, they
can identify the force pairs and the objects on which they act, and state the
magnitude and direction of each force.
b) Students should be able to apply Newton’s Third Law in analyzing the force of
contact between two objects that accelerate together along a horizontal or vertical
line, or between two surfaces that slide across one another.
c) Students should know that the tension is constant in a light string that passes over a
massless pulley and should be able to use this fact in analyzing the motion of a
system of two objects joined by a string.
d) Students should be able to solve problems in which application of Newton’s laws
leads to two or three simultaneous linear equations involving unknown forces or
accelerations.
C. Work, energy, power
1. Work and the work-energy theorem
a) Students should understand the definition of work, including when it is positive,
negative, or zero, so they can:
(1) Calculate the work done by a specified constant force on an object that
undergoes a specified displacement.
(2) Relate the work done by a force to the area under a graph of force as a function
of position, and calculate this work in the case where the force is a linear
function of position.
(3) Use integration to calculate the work performed by a force F(x) on an object
that undergoes a specified displacement in one dimension.
(4) Use the scalar product operation to calculate the work performed by a specified
constant force F on an object that undergoes a displacement in a plane.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
AP Course
B
C
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Objectives for the AP® Physics Courses
b) Students should understand and be able to apply the work-energy theorem, so they
can:
(1) Calculate the change in kinetic energy or speed that results from performing a
specified amount of work on an object.
(2) Calculate the work performed by the net force, or by each of the forces that
make up the net force, on an object that undergoes a specified change in speed
or kinetic energy.
(3) Apply the theorem to determine the change in an object’s kinetic energy and
speed that results from the application of specified forces, or to determine the
force that is required in order to bring an object to rest in a specified distance.
2. Forces and potential energy
a) Students should understand the concept of a conservative force, so they can:
(1) State alternative definitions of “conservative force” and explain why these
definitions are equivalent.
(2) Describe examples of conservative forces and non-conservative forces.
b) Students should understand the concept of potential energy, so they can:
(1) State the general relation between force and potential energy, and explain why
potential energy can be associated only with conservative forces.
(2) Calculate a potential energy function associated with a specified onedimensional force F(x).
(3) Calculate the magnitude and direction of a one-dimensional force when given
the potential energy function U(x) for the force.
(4) Write an expression for the force exerted by an ideal spring and for the potential
energy of a stretched or compressed spring.
(5) Calculate the potential energy of one or more objects in a uniform gravitational
field.
3. Conservation of energy
a) Students should understand the concepts of mechanical energy and of total energy,
so they can:
(1) State and apply the relation between the work performed on an object by nonconservative forces and the change in an object’s mechanical energy.
(2) Describe and identify situations in which mechanical energy is converted to
other forms of energy.
(3) Analyze situations in which an object’s mechanical energy is changed by
friction or by a specified externally applied force.
b) Students should understand conservation of energy, so they can:
(1) Identify situations in which mechanical energy is or is not conserved.
(2) Apply conservation of energy in analyzing the motion of systems of connected
objects, such as an Atwood’s machine.
(3) Apply conservation of energy in analyzing the motion of objects that move
under the influence of springs.
(4) Apply conservation of energy in analyzing the motion of objects that move
under the influence of other non-constant one-dimensional forces.
c) Students should be able to recognize and solve problems that call for application
both of conservation of energy and Newton’s Laws.
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© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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Objectives for the AP® Physics Courses
4. Power
Students should understand the definition of power, so they can:
a) Calculate the power required to maintain the motion of an object with constant
acceleration (e.g., to move an object along a level surface, to raise an object at a
constant rate, or to overcome friction for an object that is moving at a constant
speed).
b) Calculate the work performed by a force that supplies constant power, or the
average power supplied by a force that performs a specified amount of work.
D. Systems of particles, linear momentum
1. Center of mass
a) Students should understand the technique for finding center of mass, so they can:
(1) Identify by inspection the center of mass of a symmetrical object.
(2) Locate the center of mass of a system consisting of two such objects.
(3) Use integration to find the center of mass of a thin rod of non-uniform density.
b) Students should be able to understand and apply the relation between center-ofmass velocity and linear momentum, and between center-of-mass acceleration and
net external force for a system of particles.
c) Students should be able to define center of gravity and to use this concept to
express the gravitational potential energy of a rigid object in terms of the position
of its center of mass.
2. Impulse and momentum
Students should understand impulse and linear momentum, so they can:
a) Relate mass, velocity, and linear momentum for a moving object, and calculate the
total linear momentum of a system of objects.
b) Relate impulse to the change in linear momentum and the average force acting on
an object.
c) State and apply the relations between linear momentum and center-of-mass motion
for a system of particles.
d) Calculate the area under a force versus time graph and relate it to the change in
momentum of an object.
e) Calculate the change in momentum of an object given a function F (t ) for the net
force acting on the object.
3. Conservation of linear momentum, collisions
a) Students should understand linear momentum conservation, so they can:
(1) Explain how linear momentum conservation follows as a consequence of
Newton’s Third Law for an isolated system.
(2) Identify situations in which linear momentum, or a component of the linear
momentum vector, is conserved.
(3) Apply linear momentum conservation to one-dimensional elastic and inelastic
collisions and two-dimensional completely inelastic collisions.
(4) Apply linear momentum conservation to two-dimensional elastic and inelastic
collisions.
(5) Analyze situations in which two or more objects are pushed apart by a spring or
other agency, and calculate how much energy is released in such a process.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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b) Students should understand frames of reference, so they can:
(1) Analyze the uniform motion of an object relative to a moving medium such as a
flowing stream.
(2) Analyze the motion of particles relative to a frame of reference that is
accelerating horizontally or vertically at a uniform rate.
E. Circular motion and rotation
1. Uniform circular motion
Students should understand the uniform circular motion of a particle, so they can:
a) Relate the radius of the circle and the speed or rate of revolution of the particle to
the magnitude of the centripetal acceleration.
b) Describe the direction of the particle’s velocity and acceleration at any instant
during the motion.
c) Determine the components of the velocity and acceleration vectors at any instant,
and sketch or identify graphs of these quantities.
d) Analyze situations in which an object moves with specified acceleration under the
influence of one or more forces so they can determine the magnitude and direction
of the net force, or of one of the forces that makes up the net force, in situations
such as the following:
(1) Motion in a horizontal circle (e.g., mass on a rotating merry-go-round, or car
rounding a banked curve).
(2) Motion in a vertical circle (e.g., mass swinging on the end of a string, cart
rolling down a curved track, rider on a Ferris wheel).
2. Torque and rotational statics
a) Students should understand the concept of torque, so they can:
(1) Calculate the magnitude and direction of the torque associated with a given
force.
(2) Calculate the torque on a rigid object due to gravity.
b) Students should be able to analyze problems in statics, so they can:
(1) State the conditions for translational and rotational equilibrium of a rigid object.
(2) Apply these conditions in analyzing the equilibrium of a rigid object under the
combined influence of a number of coplanar forces applied at different
locations.
c) Students should develop a qualitative understanding of rotational inertia, so they
can:
(1) Determine by inspection which of a set of symmetrical objects of equal mass
has the greatest rotational inertia.
(2) Determine by what factor an object’s rotational inertia changes if all its
dimensions are increased by the same factor.
d) Students should develop skill in computing rotational inertia so they can find the
rotational inertia of:
(1) A collection of point masses lying in a plane about an axis perpendicular to the
plane.
(2) A thin rod of uniform density, about an arbitrary axis perpendicular to the rod.
(3) A thin cylindrical shell about its axis, or an object that may be viewed as being
made up of coaxial shells.
e) Students should be able to state and apply the parallel-axis theorem.
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3. Rotational kinematics and dynamics
a) Students should understand the analogy between translational and rotational
kinematics so they can write and apply relations among the angular acceleration,
angular velocity, and angular displacement of an object that rotates about a fixed
axis with constant angular acceleration.
b) Students should be able to use the right-hand rule to associate an angular velocity
vector with a rotating object.
c) Students should understand the dynamics of fixed-axis rotation, so they can:
(1) Describe in detail the analogy between fixed-axis rotation and straight-line
translation.
(2) Determine the angular acceleration with which a rigid object is accelerated
about a fixed axis when subjected to a specified external torque or force.
(3) Determine the radial and tangential acceleration of a point on a rigid object.
(4) Apply conservation of energy to problems of fixed-axis rotation.
(5) Analyze problems involving strings and massive pulleys.
d) Students should understand the motion of a rigid object along a surface, so they
can:
(1) Write down, justify, and apply the relation between linear and angular velocity,
or between linear and angular acceleration, for an object of circular crosssection that rolls without slipping along a fixed plane, and determine the
velocity and acceleration of an arbitrary point on such an object.
(2) Apply the equations of translational and rotational motion simultaneously in
analyzing rolling with slipping.
(3) Calculate the total kinetic energy of an object that is undergoing both
translational and rotational motion, and apply energy conservation in analyzing
such motion.
4. Angular momentum and its conservation
a) Students should be able to use the vector product and the right-hand rule, so they
can:
(1) Calculate the torque of a specified force about an arbitrary origin.
(2) Calculate the angular momentum vector for a moving particle.
(3) Calculate the angular momentum vector for a rotating rigid object in simple
cases where this vector lies parallel to the angular velocity vector.
b) Students should understand angular momentum conservation, so they can:
(1) Recognize the conditions under which the law of conservation is applicable and
relate this law to one- and two-particle systems such as satellite orbits.
(2) State the relation between net external torque and angular momentum, and
identify situations in which angular momentum is conserved.
(3) Analyze problems in which the moment of inertia of an object is changed as it
rotates freely about a fixed axis.
(4) Analyze a collision between a moving particle and a rigid object that can rotate
about a fixed axis or about its center of mass.
F. Oscillations and Gravitation
1. Simple harmonic motion (dynamics and energy relationships)
Students should understand simple harmonic motion, so they can:
a) Sketch or identify a graph of displacement as a function of time, and determine
from such a graph the amplitude, period and frequency of the motion.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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b) Write down an appropriate expression for displacement of the form A sin wt or
A cos wt to describe the motion.
c) Find an expression for velocity as a function of time.
d) State the relations between acceleration, velocity and displacement, and identify
points in the motion where these quantities are zero or achieve their greatest
positive and negative values.
e) State and apply the relation between frequency and period.
f) Recognize that a system that obeys a differential equation of the form
d 2 x dt 2 = - w 2 x must execute simple harmonic motion, and determine the
frequency and period of such motion.
g) State how the total energy of an oscillating system depends on the amplitude of the
motion, sketch, or identify a graph of kinetic or potential energy as a function of
time, and identify points in the motion where this energy is all potential or all
kinetic.
h) Calculate the kinetic and potential energies of an oscillating system as functions of
time, sketch or identify graphs of these functions, and prove that the sum of kinetic
and potential energy is constant.
i) Calculate the maximum displacement or velocity of a particle that moves in simple
harmonic motion with specified initial position and velocity.
j) Develop a qualitative understanding of resonance so they can identify situations in
which a system will resonate in response to a sinusoidal external force.
2. Mass on a spring
Students should be able to apply their knowledge of simple harmonic motion to the
case of a mass on a spring, so they can:
a) Derive the expression for the period of oscillation of a mass on a spring.
b) Apply the expression for the period of oscillation of a mass on a spring.
c) Analyze problems in which a mass hangs from a spring and oscillates vertically.
d) Analyze problems in which a mass attached to a spring oscillates horizontally.
e) Determine the period of oscillation for systems involving series or parallel
combinations of identical springs, or springs of differing lengths.
3. Pendulum and other oscillations
Students should be able to apply their knowledge of simple harmonic motion to the
case of a pendulum, so they can:
a) Derive the expression for the period of a simple pendulum.
b) Apply the expression for the period of a simple pendulum.
c) State what approximation must be made in deriving the period.
d) Analyze the motion of a torsional pendulum or physical pendulum in order to
determine the period of small oscillations.
4. Newton’s law of gravity
Students should know Newton’s Law of Universal Gravitation, so they can:
a) Determine the force that one spherically symmetrical mass exerts on another.
b) Determine the strength of the gravitational field at a specified point outside a
spherically symmetrical mass.
c) Describe the gravitational force inside and outside a uniform sphere, and calculate
how the field at the surface depends on the radius and density of the sphere.
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5. Orbits of planets and satellites
Students should understand the motion of an object in orbit under the influence of
gravitational forces, so they can:
a) For a circular orbit:
(1) Recognize that the motion does not depend on the object’s mass; describe
qualitatively how the velocity, period of revolution, and centripetal acceleration
depend upon the radius of the orbit; and derive expressions for the velocity and
period of revolution in such an orbit.
(2) Derive Kepler’s Third Law for the case of circular orbits.
(3) Derive and apply the relations among kinetic energy, potential energy, and total
energy for such an orbit.
b) For a general orbit:
(1) State Kepler’s three laws of planetary motion and use them to describe in
qualitative terms the motion of an object in an elliptical orbit.
(2) Apply conservation of angular momentum to determine the velocity and radial
distance at any point in the orbit.
(3) Apply angular momentum conservation and energy conservation to relate the
speeds of an object at the two extremes of an elliptical orbit.
(4) Apply energy conservation in analyzing the motion of an object that is
projected straight up from a planet’s surface or that is projected directly toward
the planet from far above the surface.
II. FLUID MECHANICS AND THERMAL PHYSICS
A. Fluid Mechanics
1. Hydrostatic pressure
Students should understand the concept of pressure as it applies to fluids, so they
can:
a) Apply the relationship between pressure, force, and area.
b) Apply the principle that a fluid exerts pressure in all directions.
c) Apply the principle that a fluid at rest exerts pressure perpendicular to any surface
that it contacts.
d) Determine locations of equal pressure in a fluid.
e) Determine the values of absolute and gauge pressure for a particular situation.
f) Apply the relationship between pressure and depth in a liquid, DP = r g Dh .
2. Buoyancy
Students should understand the concept of buoyancy, so they can:
a) Determine the forces on an object immersed partly or completely in a liquid.
b) Apply Archimedes’ principle to determine buoyant forces and densities of solids
and liquids.
3. Fluid flow continuity
Students should understand the equation of continuity so that they can apply it to
fluids in motion.
4. Bernoulli’s equation
Students should understand Bernoulli’s equation so that they can apply it to fluids in
motion.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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B. Temperature and heat
1. Mechanical equivalent of heat
Students should understand the “mechanical equivalent of heat” so they can
determine how much heat can be produced by the performance of a specified
quantity of mechanical work.
2. Heat transfer and thermal expansion
Students should understand heat transfer and thermal expansion, so they can:
a) Calculate how the flow of heat through a slab of material is affected by changes in
the thickness or area of the slab, or the temperature difference between the two
faces of the slab.
b) Analyze what happens to the size and shape of an object when it is heated.
c) Analyze qualitatively the effects of conduction, radiation, and convection in
thermal processes.
C. Kinetic theory and thermodynamics
1. Ideal gases
a) Students should understand the kinetic theory model of an ideal gas, so they can:
(1) State the assumptions of the model.
(2) State the connection between temperature and mean translational kinetic
energy, and apply it to determine the mean speed of gas molecules as a function
of their mass and the temperature of the gas.
(3) State the relationship among Avogadro’s number, Boltzmann’s constant, and
the gas constant R, and express the energy of a mole of a monatomic ideal gas
as a function of its temperature.
(4) Explain qualitatively how the model explains the pressure of a gas in terms of
collisions with the container walls, and explain how the model predicts that, for
fixed volume, pressure must be proportional to temperature.
b) Students should know how to apply the ideal gas law and thermodynamic
principles, so they can:
(1) Relate the pressure and volume of a gas during an isothermal expansion or
compression.
(2) Relate the pressure and temperature of a gas during constant-volume heating or
cooling, or the volume and temperature during constant-pressure heating or
cooling.
(3) Calculate the work performed on or by a gas during an expansion or
compression at constant pressure.
(4) Understand the process of adiabatic expansion or compression of a gas.
(5) Identify or sketch on a PV diagram the curves that represent each of the above
processes.
2. Laws of thermodynamics
a) Students should know how to apply the first law of thermodynamics, so they can:
(1) Relate the heat absorbed by a gas, the work performed by the gas, and the
internal energy change of the gas for any of the processes above.
(2) Relate the work performed by a gas in a cyclic process to the area enclosed by a
curve on a PV diagram.
b) Students should understand the second law of thermodynamics, the concept of
entropy, and heat engines and the Carnot cycle, so they can:
(1) Determine whether entropy will increase, decrease, or remain the same during a
particular situation.
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Objectives for the AP® Physics Courses
(2) Compute the maximum possible efficiency of a heat engine operating between
two given temperatures.
(3) Compute the actual efficiency of a heat engine.
(4) Relate the heats exchanged at each thermal reservoir in a Carnot cycle to the
temperatures of the reservoirs.
III. ELECTRICITY AND MAGNETISM
A. Electrostatics
1. Charge and Coulomb’s Law
a) Students should understand the concept of electric charge, so they can:
(1) Describe the types of charge and the attraction and repulsion of charges.
(2) Describe polarization and induced charges.
b) Students should understand Coulomb’s Law and the principle of superposition, so
they can:
(1) Calculate the magnitude and direction of the force on a positive or negative
charge due to other specified point charges.
(2) Analyze the motion of a particle of specified charge and mass under the
influence of an electrostatic force.
2. Electric field and electric potential (including point charges)
a) Students should understand the concept of electric field, so they can:
(1) Define it in terms of the force on a test charge.
(2) Describe and calculate the electric field of a single point charge.
(3) Calculate the magnitude and direction of the electric field produced by two or
more point charges.
(4) Calculate the magnitude and direction of the force on a positive or negative
charge placed in a specified field.
(5) Interpret an electric field diagram.
(6) Analyze the motion of a particle of specified charge and mass in a uniform
electric field.
b) Students should understand the concept of electric potential, so they can:
(1) Determine the electric potential in the vicinity of one or more point charges.
(2) Calculate the electrical work done on a charge or use conservation of energy to
determine the speed of a charge that moves through a specified potential
difference.
(3) Determine the direction and approximate magnitude of the electric field at
various positions given a sketch of equipotentials.
(4) Calculate the potential difference between two points in a uniform electric field,
and state which point is at the higher potential.
(5) Calculate how much work is required to move a test charge from one location
to another in the field of fixed point charges.
(6) Calculate the electrostatic potential energy of a system of two or more point
charges, and calculate how much work is required to establish the charge
system.
(7) Use integration to determine electric potential difference between two points on
a line, given electric field strength as a function of position along that line.
(8) State the general relationship between field and potential, and define and apply
the concept of a conservative electric field.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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3. Gauss’s law
a) Students should understand the relationship between electric field and electric flux,
so they can:
(1) Calculate the flux of an electric field through an arbitrary surface or of a field
uniform in magnitude over a Gaussian surface and perpendicular to it.
(2) Calculate the flux of the electric field through a rectangle when the field is
perpendicular to the rectangle and a function of one coordinate only.
(3) State and apply the relationship between flux and lines of force.
b) Students should understand Gauss’s Law, so they can:
(1) State the law in integral form, and apply it qualitatively to relate flux and
electric charge for a specified surface.
(2) Apply the law, along with symmetry arguments, to determine the electric field
for a planar, spherical, or cylindrically symmetric charge distribution.
(3) Apply the law to determine the charge density or total charge on a surface in
terms of the electric field near the surface.
4. Fields and potentials of other charge distributions
a) Students should be able to use the principle of superposition to calculate by
integration:
(1) The electric field of a straight, uniformly charged wire.
(2) The electric field and potential on the axis of a thin ring of charge, or at the
center of a circular arc of charge.
(3) The electric potential on the axis of a uniformly charged disk.
b) Students should know the fields of highly symmetric charge distributions, so they
can:
(1) Identify situations in which the direction of the electric field produced by a
charge distribution can be deduced from symmetry considerations.
(2) Describe qualitatively the patterns and variation with distance of the electric
field of:
(a) Oppositely-charged parallel plates.
(b) A long, uniformly-charged wire, or thin cylindrical or spherical shell.
(3) Use superposition to determine the fields of parallel charged planes, coaxial
cylinders, or concentric spheres.
(4) Derive expressions for electric potential as a function of position in the above
cases.
B. Conductors, capacitors, dielectrics
1. Electrostatics with conductors
a) Students should understand the nature of electric fields in and around conductors,
so they can:
(1) Explain the mechanics responsible for the absence of electric field inside a
conductor, and know that all excess charge must reside on the surface of the
conductor.
(2) Explain why a conductor must be an equipotential, and apply this principle in
analyzing what happens when conductors are connected by wires.
(3) Show that all excess charge on a conductor must reside on its surface and that
the field outside the conductor must be perpendicular to the surface.
b) Students should be able to describe and sketch a graph of the electric field and
potential inside and outside a charged conducting sphere.
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c) Students should understand induced charge and electrostatic shielding, so they can:
(1) Describe the process of charging by induction.
(2) Explain why a neutral conductor is attracted to a charged object.
(3) Explain why there can be no electric field in a charge-free region completely
surrounded by a single conductor, and recognize consequences of this result.
(4) Explain why the electric field outside a closed conducting surface cannot
depend on the precise location of charge in the space enclosed by the conductor,
and identify consequences of this result.
2. Capacitors
a) Students should understand the definition and function of capacitance, so they can:
(1) Relate stored charge and voltage for a capacitor.
(2) Relate voltage, charge, and stored energy for a capacitor.
(3) Recognize situations in which energy stored in a capacitor is converted to other
forms.
b) Students should understand the physics of the parallel-plate capacitor, so they can:
(1) Describe the electric field inside the capacitor, and relate the strength of this
field to the potential difference between the plates and the plate separation.
(2) Relate the electric field to the density of the charge on the plates.
(3) Derive an expression for the capacitance of a parallel-plate capacitor.
(4) Determine how changes in dimension will affect the value of the capacitance.
(5) Derive and apply expressions for the energy stored in a parallel-plate capacitor
and for the energy density in the field between the plates.
(6) Analyze situations in which capacitor plates are moved apart or moved closer
together, or in which a conducting slab is inserted between capacitor plates,
either with a battery connected between the plates or with the charge on the
plates held fixed.
c) Students should understand cylindrical and spherical capacitors, so they can:
(1) Describe the electric field inside each.
(2) Derive an expression for the capacitance of each.
3. Dielectrics
Students should understand the behavior of dielectrics, so they can:
a) Describe how the insertion of a dielectric between the plates of a charged parallelplate capacitor affects its capacitance and the field strength and voltage between the
plates.
b) Analyze situations in which a dielectric slab is inserted between the plates of a
capacitor.
C. Electric circuits
1. Current, resistance, power
a) Students should understand the definition of electric current, so they can relate the
magnitude and direction of the current to the rate of flow of positive and negative
charge.
b) Students should understand conductivity, resistivity and resistance, so they can:
(1) Relate current and voltage for a resistor.
(2) Write the relationship between electric field strength and current density in a
conductor, and describe, in terms of the drift velocity of electrons, why such a
relationship is plausible.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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(3) Describe how the resistance of a resistor depends upon its length and crosssectional area, and apply this result in comparing current flow in resistors of
different material or different geometry.
(4) Derive an expression for the resistance of a resistor of uniform cross-section in
terms of its dimensions and the resistivity of the material from which it is
constructed.
(5) Derive expressions that relate the current, voltage and resistance to the rate at
which heat is produced when current passes through a resistor.
(6) Apply the relationships for the rate of heat production in a resistor.
2. Steady-state direct current circuits with batteries and resistors only
a) Students should understand the behavior of series and parallel combinations of
resistors, so they can:
(1) Identify on a circuit diagram whether resistors are in series or in parallel.
(2) Determine the ratio of the voltages across resistors connected in series or the
ratio of the currents through resistors connected in parallel.
(3) Calculate the equivalent resistance of a network of resistors that can be broken
down into series and parallel combinations.
(4) Calculate the voltage, current, and power dissipation for any resistor in such a
network of resistors connected to a single power supply.
(5) Design a simple series-parallel circuit that produces a given current through and
potential difference across one specified component, and draw a diagram for the
circuit using conventional symbols.
b) Students should understand the properties of ideal and real batteries, so they can:
(1) Calculate the terminal voltage of a battery of specified emf and internal
resistance from which a known current is flowing.
(2) Calculate the rate at which a battery is supplying energy to a circuit or is being
charged up by a circuit.
c) Students should be able to apply Ohm’s law and Kirchhoff’s rules to direct-current
circuits, in order to:
(1) Determine a single unknown current, voltage, or resistance.
(2) Set up and solve simultaneous equations to determine two unknown currents.
d) Students should understand the properties of voltmeters and ammeters, so they can:
(1) State whether the resistance of each is high or low.
(2) Identify or show correct methods of connecting meters into circuits in order to
measure voltage or current.
(3) Assess qualitatively the effect of finite meter resistance on a circuit into which
these meters are connected.
3. Capacitors in circuits
a) Students should understand the t = 0 and steady-state behavior of capacitors
connected in series or in parallel, so they can:
(1) Calculate the equivalent capacitance of a series or parallel combination.
(2) Describe how stored charge is divided between capacitors connected in parallel.
(3) Determine the ratio of voltages for capacitors connected in series.
(4) Calculate the voltage or stored charge, under steady-state conditions, for a
capacitor connected to a circuit consisting of a battery and resistors.
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Objectives for the AP® Physics Courses
b) Students should understand the discharging or charging of a capacitor through a
resistor, so they can:
(1) Calculate and interpret the time constant of the circuit.
(2) Sketch or identify graphs of stored charge or voltage for the capacitor, or of
current or voltage for the resistor, and indicate on the graph the significance of
the time constant.
(3) Write expressions to describe the time dependence of the stored charge or
voltage for the capacitor, or of the current or voltage for the resistor.
(4) Analyze the behavior of circuits containing several capacitors and resistors,
including analyzing or sketching graphs that correctly indicate how voltages and
currents vary with time.
D. Magnetic Fields
1. Forces on moving charges in magnetic fields
Students should understand the force experienced by a charged particle in a magnetic
field, so they can:
a) Calculate the magnitude and direction of the force in terms of q, v, and B, and
explain why the magnetic force can perform no work.
b) Deduce the direction of a magnetic field from information about the forces
experienced by charged particles moving through that field.
c) Describe the paths of charged particles moving in uniform magnetic fields.
d) Derive and apply the formula for the radius of the circular path of a charge that
moves perpendicular to a uniform magnetic field.
e) Describe under what conditions particles will move with constant velocity through
crossed electric and magnetic fields.
2. Forces on current-carrying wires in magnetic fields
Students should understand the force exerted on a current-carrying wire in a magnetic
field, so they can:
a) Calculate the magnitude and direction of the force on a straight segment of currentcarrying wire in a uniform magnetic field.
b) Indicate the direction of magnetic forces on a current-carrying loop of wire in a
magnetic field, and determine how the loop will tend to rotate as a consequence of
these forces.
c) Calculate the magnitude and direction of the torque experienced by a rectangular
loop of wire carrying a current in a magnetic field.
3. Fields of long current-carrying wires
Students should understand the magnetic field produced by a long straight currentcarrying wire, so they can:
a) Calculate the magnitude and direction of the field at a point in the vicinity of such a
wire.
b) Use superposition to determine the magnetic field produced by two long wires.
c) Calculate the force of attraction or repulsion between two long current-carrying
wires.
4. Biot-Savart law and Ampere’s law
a) Students should understand the Biot-Savart Law, so they can:
(1) Deduce the magnitude and direction of the contribution to the magnetic field
made by a short straight segment of current-carrying wire.
(2) Derive and apply the expression for the magnitude of B on the axis of a circular
loop of current.
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b) Students should understand the statement and application of Ampere’s Law in
integral form, so they can:
(1) State the law precisely.
(2) Use Ampere’s law, plus symmetry arguments and the right-hand rule, to relate
magnetic field strength to current for planar or cylindrical symmetries.
c) Students should be able to apply the superposition principle so they can determine
the magnetic field produced by combinations of the configurations listed above.
E. Electromagnetism
1. Electromagnetic induction (including Faraday’s law and Lenz’s law)
a) Students should understand the concept of magnetic flux, so they can:
(1) Calculate the flux of a uniform magnetic field through a loop of arbitrary
orientation.
(2) Use integration to calculate the flux of a non-uniform magnetic field, whose
magnitude is a function of one coordinate, through a rectangular loop
perpendicular to the field.
b) Students should understand Faraday’s law and Lenz’s law, so they can:
(1) Recognize situations in which changing flux through a loop will cause an
induced emf or current in the loop.
(2) Calculate the magnitude and direction of the induced emf and current in a loop
of wire or a conducting bar under the following conditions:
(a) The magnitude of a related quantity such as magnetic field or area of the
loop is changing at a constant rate.
(b) The magnitude of a related quantity such as magnetic field or area of the
loop is a specified non-linear function of time.
c) Students should be able to analyze the forces that act on induced currents so they
can determine the mechanical consequences of those forces.
2. Inductance (including LR and LC circuits)
a) Students should understand the concept of inductance, so they can:
(1) Calculate the magnitude and sense of the emf in an inductor through which a
specified changing current is flowing.
(2) Derive and apply the expression for the self-inductance of a long solenoid.
b) Students should understand the transient and steady state behavior of DC circuits
containing resistors and inductors, so they can:
(1) Apply Kirchhoff’s rules to a simple LR series circuit to obtain a differential
equation for the current as a function of time.
(2) Solve the differential equation obtained in (1) for the current as a function of
time through the battery, using separation of variables.
(3) Calculate the initial transient currents and final steady state currents through
any part of a simple series and parallel circuit containing an inductor and one or
more resistors.
(4) Sketch graphs of the current through or voltage across the resistors or inductor
in a simple series and parallel circuit.
(5) Calculate the rate of change of current in the inductor as a function of time.
(6) Calculate the energy stored in an inductor that has a steady current flowing
through it.
3. Maxwell’s equations
Students should be familiar with Maxwell’s equations so they can associate each
equation with its implications.
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IV. WAVES AND OPTICS
A. Wave motion (including sound)
1. Traveling waves
Students should understand the description of traveling waves, so they can:
a) Sketch or identify graphs that represent traveling waves and determine the
amplitude, wavelength, and frequency of a wave from such a graph.
b) Apply the relation among wavelength, frequency, and velocity for a wave.
c) Understand qualitatively the Doppler effect for sound in order to explain why there
is a frequency shift in both the moving-source and moving-observer case.
d) Describe reflection of a wave from the fixed or free end of a string.
e) Describe qualitatively what factors determine the speed of waves on a string and
the speed of sound.
2. Wave propagation
a) Students should understand the difference between transverse and longitudinal
waves, and be able to explain qualitatively why transverse waves can exhibit
polarization.
b) Students should understand the inverse-square law, so they can calculate the
intensity of waves at a given distance from a source of specified power and
compare the intensities at different distances from the source.
3. Standing waves
Students should understand the physics of standing waves, so they can:
a) Sketch possible standing wave modes for a stretched string that is fixed at both
ends, and determine the amplitude, wavelength, and frequency of such standing
waves.
b) Describe possible standing sound waves in a pipe that has either open or closed
ends, and determine the wavelength and frequency of such standing waves.
4. Superposition
Students should understand the principle of superposition, so they can apply it to
traveling waves moving in opposite directions, and describe how a standing wave
may be formed by superposition.
B. Physical optics
1. Interference and diffraction
Students should understand the interference and diffraction of waves, so they can:
a) Apply the principles of interference to coherent sources in order to:
(1) Describe the conditions under which the waves reaching an observation point
from two or more sources will all interfere constructively, or under which the
waves from two sources will interfere destructively.
(2) Determine locations of interference maxima or minima for two sources or
determine the frequencies or wavelengths that can lead to constructive or
destructive interference at a certain point.
(3) Relate the amplitude produced by two or more sources that interfere
constructively to the amplitude and intensity produced by a single source.
b) Apply the principles of interference and diffraction to waves that pass through a
single or double slit or through a diffraction grating, so they can:
(1) Sketch or identify the intensity pattern that results when monochromatic waves
pass through a single slit and fall on a distant screen, and describe how this
pattern will change if the slit width or the wavelength of the waves is changed.
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(2) Calculate, for a single-slit pattern, the angles or the positions on a distant screen
where the intensity is zero.
(3) Sketch or identify the intensity pattern that results when monochromatic waves
pass through a double slit, and identify which features of the pattern result from
single-slit diffraction and which from two-slit interference.
(4) Calculate, for a two-slit interference pattern, the angles or the positions on a
distant screen at which intensity maxima or minima occur.
(5) Describe or identify the interference pattern formed by a diffraction grating,
calculate the location of intensity maxima, and explain qualitatively why a
multiple-slit grating is better than a two-slit grating for making accurate
determinations of wavelength.
c) Apply the principles of interference to light reflected by thin films, so they can:
(1) State under what conditions a phase reversal occurs when light is reflected from
the interface between two media of different indices of refraction.
(2) Determine whether rays of monochromatic light reflected perpendicularly from
two such interfaces will interfere constructively or destructively, and thereby
account for Newton’s rings and similar phenomena, and explain how glass may
be coated to minimize reflection of visible light.
2. Dispersion of light and the electromagnetic spectrum
Students should understand dispersion and the electromagnetic spectrum, so they can:
a) Relate a variation of index of refraction with frequency to a variation in refraction.
b) Know the names associated with electromagnetic radiation and be able to arrange
in order of increasing wavelength the following: visible light of various colors,
ultraviolet light, infrared light, radio waves, x-rays, and gamma rays.
C. Geometric optics
1. Reflection and refraction
Students should understand the principles of reflection and refraction, so they can:
a) Determine how the speed and wavelength of light change when light passes from
one medium into another.
b) Show on a diagram the directions of reflected and refracted rays.
c) Use Snell’s Law to relate the directions of the incident ray and the refracted ray,
and the indices of refraction of the media.
d) Identify conditions under which total internal reflection will occur.
2. Mirrors
Students should understand image formation by plane or spherical mirrors, so they
can:
a) Locate by ray tracing the image of an object formed by a plane mirror, and
determine whether the image is real or virtual, upright or inverted, enlarged or
reduced in size.
b) Relate the focal point of a spherical mirror to its center of curvature.
c) Locate by ray tracing the image of a real object, given a diagram of a mirror with
the focal point shown, and determine whether the image is real or virtual, upright or
inverted, enlarged or reduced in size.
d) Use the mirror equation to relate the object distance, image distance, and focal
length for a lens, and determine the image size in terms of the object size.
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Objectives for the AP® Physics Courses
3. Lenses
Students should understand image formation by converging or diverging lenses, so
they can:
a) Determine whether the focal length of a lens is increased or decreased as a result of
a change in the curvature of its surfaces, or in the index of refraction of the material
of which the lens is made, or the medium in which it is immersed.
b) Determine by ray tracing the location of the image of a real object located inside or
outside the focal point of the lens, and state whether the resulting image is upright
or inverted, real or virtual.
c) Use the thin lens equation to relate the object distance, image distance and focal
length for a lens, and determine the image size in terms of the object size.
d) Analyze simple situations in which the image formed by one lens serves as the
object for another lens.
V. ATOMIC AND NUCLEAR PHYSICS
A. Atomic physics and quantum effects
1. Photons, the photoelectric effect, Compton scattering, x-rays
a) Students should know the properties of photons, so they can:
(1) Relate the energy of a photon in joules or electron-volts to its wavelength or
frequency.
(2) Relate the linear momentum of a photon to its energy or wavelength, and apply
linear momentum conservation to simple processes involving the emission,
absorption, or reflection of photons.
(3) Calculate the number of photons per second emitted by a monochromatic
source of specific wavelength and power.
b) Students should understand the photoelectric effect, so they can:
(1) Describe a typical photoelectric-effect experiment, and explain what
experimental observations provide evidence for the photon nature of light.
(2) Describe qualitatively how the number of photoelectrons and their maximum
kinetic energy depend on the wavelength and intensity of the light striking the
surface, and account for this dependence in terms of a photon model of light.
(3) Determine the maximum kinetic energy of photoelectrons ejected by photons of
one energy or wavelength, when given the maximum kinetic energy of
photoelectrons for a different photon energy or wavelength.
(4) Sketch or identify a graph of stopping potential versus frequency for a
photoelectric-effect experiment, determine from such a graph the threshold
frequency and work function, and calculate an approximate value of h/e.
c) Students should understand Compton scattering, so they can:
(1) Describe Compton’s experiment, and state what results were observed and by
what sort of analysis these results may be explained.
(2) Account qualitatively for the increase of photon wavelength that is observed,
and explain the significance of the Compton wavelength.
d) Students should understand the nature and production of x-rays, so they can
calculate the shortest wavelength of x-rays that may be produced by electrons
accelerated through a specified voltage.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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2. Atomic energy levels
Students should understand the concept of energy levels for atoms, so they can:
a) Calculate the energy or wavelength of the photon emitted or absorbed in a
transition between specified levels, or the energy or wavelength required to ionize
an atom.
b) Explain qualitatively the origin of emission or absorption spectra of gases.
c) Calculate the wavelength or energy for a single-step transition between levels,
given the wavelengths or energies of photons emitted or absorbed in a two-step
transition between the same levels.
d) Draw a diagram to depict the energy levels of an atom when given an expression
for these levels, and explain how this diagram accounts for the various lines in the
atomic spectrum.
3. Wave-particle duality
Students should understand the concept of de Broglie wavelength, so they can:
a) Calculate the wavelength of a particle as a function of its momentum.
b) Describe the Davisson-Germer experiment, and explain how it provides evidence
for the wave nature of electrons.
B. Nuclear Physics
1. Nuclear reactions (including conservation of mass number and charge)
a) Students should understand the significance of the mass number and charge of
nuclei, so they can:
(1) Interpret symbols for nuclei that indicate these quantities.
(2) Use conservation of mass number and charge to complete nuclear reactions.
(3) Determine the mass number and charge of a nucleus after it has undergone
specified decay processes.
b) Students should know the nature of the nuclear force, so they can compare its
strength and range with those of the electromagnetic force.
c) Students should understand nuclear fission, so they can describe a typical neutroninduced fission and explain why a chain reaction is possible.
2. Mass-energy equivalence
Students should understand the relationship between mass and energy (mass-energy
equivalence), so they can:
a) Qualitatively relate the energy released in nuclear processes to the change in mass.
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b) Apply the relationship DE = ( Dm) c in analyzing nuclear processes.
LABORATORY AND EXPERIMENTAL SITUATIONS
These objectives overlay the content objectives, and are assessed in the context of those
objectives.
1. Design experiments
Students should understand the process of designing experiments, so they can:
a) Describe the purpose of an experiment or a problem to be investigated.
b) Identify equipment needed and describe how it is to be used.
c) Draw a diagram or provide a description of an experimental setup.
d) Describe procedures to be used, including controls and measurements to be taken.
2. Observe and measure real phenomena
Students should be able to make relevant observations, and be able to take
measurements with a variety of instruments (cannot be assessed via paper-and-pencil
examinations).
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3. Analyze data
Students should understand how to analyze data, so they can:
a) Display data in graphical or tabular form.
b) Fit lines and curves to data points in graphs.
c) Perform calculations with data.
d) Make extrapolations and interpolations from data.
4. Analyze errors
Students should understand measurement and experimental error, so they can:
a) Identify sources of error and how they propagate.
b) Estimate magnitude and direction of errors.
c) Determine significant digits.
d) Identify ways to reduce error.
5. Communicate results
Students should understand how to summarize and communicate results, so they can:
a) Draw inferences and conclusions from experimental data.
b) Suggest ways to improve experiment.
c) Propose questions for further study.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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T h e E x am s
The AP Physics B Exam is 3 hours long, divided equally between a 70-question
multiple-choice section and a free-response section. The two sections are weighted
equally, and a single score is reported for the B Exam.
The free-response section will usually contain 6 or 7 questions. Examples of
possible formats are 2 questions of about 17 minutes each and 5 shorter questions
of about 11 minutes each, or 4 questions of about 17 minutes each and 2 shorter
questions of about 11 minutes each. However, future exams might include a
combination of questions of other lengths.
Each Physics C Exam is 1 hour and 30 ­minutes long. A student may take either or
both exams, and separate scores are reported for each. The time for each exam is
divided equally between a 35-question multiple-choice section and a free-response
section; the two sections are weighted equally in the ­determination of each score. The
usual format for each free-response section has been 3 questions, each taking about
15 minutes. However, future exams might include a larger number of shorter
questions.
The percentages of each exam devoted to each major category are ­specified in the
preceding pages. Departures from these percentages in the free-response section in
any given year are compensated for in the ­multiple-choice section so that the overall
topic distribution for the entire exam is achieved as closely as possible, although it
may not be reached exactly.
Some questions, particularly in the free-response sections, may involve topics from
two or more major categories. For example, a question may utilize a setting involving
principles from electricity and magnetism or atomic and nuclear physics, but parts of
the question may also involve the application of principles of mechanics to this setting,
either alone or in combination with the principles from electricity and magnetism or
atomic and nuclear physics. Such a question would not be classified uniquely according to any particular topic but would receive partial classifications by topics in
proportion to the principles needed to arrive at the answers.
On both exams the multiple-choice section emphasizes the breadth of the students’
knowledge and understanding of the basic principles of physics; the free-response
section emphasizes the application of these principles in greater depth in solving more
extended problems. In general, questions may ask students to:
• determine directions of vectors or paths of particles;
• draw or interpret diagrams;
• interpret or express physical relationships in graphical form;
• account for observed phenomena;
• interpret experimental data, including their limitations and uncertainties;
• construct and use conceptual models and explain their limitations;
• explain steps taken to arrive at a result or to predict future physical behavior;
• manipulate equations that describe physical relationships;
• obtain reasonable estimates; or
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© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
• solve problems that require the determination of physical quantities in either
numerical or symbolic form and that may require the application of single or
multiple physical concepts.
Laboratory-related questions may ask students to:
• design experiments, including identifying equipment needed and describing
how it is to be used, drawing diagrams or providing descriptions of experimental
setups, or describing procedures to be used, including controls and measurements to be taken;
• analyze data, including displaying data in graphical or tabular form, fitting lines
and curves to data points in graphs, performing calculations with data or making
extrapolations and interpolations from data;
• analyze errors, including identifying sources of errors and how they propagate,
estimating magnitude and direction of errors, determining significant digits or
identifying ways to reduce errors; or
• communicate results, including drawing inferences and conclusions from
experimental data, suggesting ways to improve experiments or proposing
questions for further study.
The free-response section of each exam is printed in a separate booklet in which each
part of a question is followed by a blank space for the student’s solution. The freeresponse section also contains a Table of Information and tables of commonly used
equations. The Table of Information, which is also printed near the front of each
multiple-choice section, includes numerical values of some physical constants and
conversion factors and states some conventions used in the exams. The equation
tables are described in greater detail in a later section.
The International System of Units (SI) is used predominantly in both exams. The
use of rulers or straightedges is permitted on the free-response sections to facilitate
the sketching of graphs or diagrams that might be required in these sections.
Since the complete exams are intended to provide the maximum information about
differences in students’ achievement in physics, students may find them more difficult
than many classroom exams. The best way for teachers to familiarize their students
with the level of difficulty is to give them actual released exams (both multiple-choice
and free-response sections) from past administrations. Information about ordering
publications is on page 81. Recent free-response sections can also be found on AP
Central, along with scoring guidelines and some sample student responses.
The Free-Response Sections — Student Presentation
Students are expected to show their work in the spaces provided for the solution for
each part of a free-response question. If they need more space, they should clearly
indicate where the work is continued or they may lose credit for it. If students make a
mistake, they may cross it out or erase it. Crossed-out work will not be scored, and
credit may be lost for incorrect work that is not crossed out.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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In scoring the free-response sections, credit for the answers depends on the quality
of the solutions and the explanations given; partial solutions may receive partial credit,
so students are advised to show all their work. Correct answers without supporting
work may lose credit. This is especially true when students are asked specifically to
justify their answers, in which case the Exam Readers are looking for some verbal or
mathematical analysis that shows how the students arrived at their answers. Also, all
final numerical answers should include appropriate units.
On the AP Physics Exams the words “justify,” “explain,” “calculate,” “what
is,” “determine,” “derive,” “sketch,” and “plot” have precise meanings.
Students should pay careful attention to these words in order to obtain maximum
credit and should avoid including irrelevant or extraneous material in their answers.
The ability to justify an answer in words shows understanding of the principles
underlying physical phenomena in addition to the ability to perform the mathematical
manipulations necessary to generate a correct answer. Students will be directed to
justify or explain their answers on many of the questions they encounter on the AP
Physics Exams. The words “justify” and “explain” indicate that the student should
support the answer with prose, equations, calculations, diagrams, or graphs. The prose
or equations may in some cases refer to fundamental ideas or relations in physics,
such as Newton’s laws, conservation of energy, Gauss’s law, or Bernoulli’s equation. In
other cases, the justification or explanation may take the form of analyzing the behavior
of an equation for large or small values of a variable in the equation.
The words “calculate,” “what is,” “determine,” and “derive” have distinct meanings
on the AP Physics Exams. “Calculate” means that a student is expected to show work
leading to a final answer, which may be algebraic but more often is numerical. “What
is” and “determine” indicate that work need not necessarily be explicitly shown to
obtain full credit. Showing work leading to answers is a good idea, as it may earn a
student partial credit in the case of an incorrect answer, but this step may be skipped
by the confident or harried student. “Derive” is more specific and indicates that the
students need to begin their solutions with one or more fundamental equations, such
as those given on the AP Physics Exam equation sheet. The final answer, usually
algebraic, is then obtained through the appropriate use of mathematics.
The words “sketch” and “plot” relate to student-produced graphs. “Sketch” means to
draw a graph that illustrates key trends in a particular relationship, such as slope,
curvature, intercept(s), or asymptote(s). Numerical scaling or specific data points are
not required in a sketch. “Plot” means to draw the data points given in the problem on
the grid provided, either using the given scale or indicating the scale and units when
none are provided.
Exam questions that require the drawing of free-body or force diagrams will direct
the students to “draw and label the forces (not components) that act on the [object]”,
where [object] is replaced by a reference specific to the question, such as “the car
when it reaches the top of the hill.” Any components that are included in the diagram
will be scored in the same way as incorrect or extraneous forces. In addition, in any
subsequent part asking for a solution that would typically make use of the diagram, the
following will be included: “If you need to draw anything other than what you have
shown in part [x] to assist in your solution, use the space below. Do NOT add anything
to the figure in part [x].” This will give students the opportunity to construct a
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working diagram showing any components that are appropriate to the solution of the
problem. This second diagram will not be scored.
Strict rules regarding significant digits are usually not applied to the scoring of
numerical answers. However, in some cases answers containing too many digits may
be penalized. In general, two to four significant digits are acceptable. Exceptions to
these guidelines usually occur when rounding makes a difference in obtaining a
reasonable answer. For example, suppose a solution requires subtracting two numbers
that should have five significant digits and that differ starting with the fourth digit
(e.g., 20.295 and 20.278). Rounding to three digits will lose the accuracy required to
determine the difference in the numbers, and some credit may be lost.
Simplification of algebraic and numerical answers is encouraged, though it should
always be balanced with students’ efficient use of exam time. Simplifying an answer
will often reveal a characteristic of the underlying physics that may be useful in a
subsequent part of the exam question. A simplified answer is the clearest way to
communicate with the professors and AP teachers who score the exams. Equivalent
answers are entitled to full credit, and the Exam Readers always evaluate unsimplified
answers for correctness. Yet, however careful the Readers are, there is always the
chance for error in their evaluations, and thus simplification may be in the students’
best interest.
Additional information about study skills and test-taking strategies can be found at
AP Central.
Calculators and Equation Tables
Policies regarding the use of calculators on the exams take into account the expansion
of the capabilities of scientific calculators, which now include not only programming
and graphing functions but also the availability of stored equations and other data. For
taking the sections of the exams in which calculators are permitted, students should
be allowed to use the calculators to which they are accustomed, except as noted
below.* On the other hand, they should not have access to information in their
­calculators that is not available to other students, if that information is needed to
answer the questions.
Calculators are not permitted on the multiple-choice sections of the
Physics B and Physics C exams. The purpose of the multiple-choice sections is to
assess the breadth of students’ knowledge and under­standing of the basic concepts
of physics. The multiple-choice questions ­emphasize conceptual understanding and
qualitative applications. However, many physical definitions and principles are quantitative by nature and can therefore be expressed as equations. The knowledge of these
basic definitions and principles, expressed as equations, is a part of the content of
physics that should be learned by physics students and will continue to be assessed in
the multiple-choice sections. However, any numeric calculations using these equations
required in the multiple-choice sections will be kept simple. Also, in some questions,
*Exceptions to calculator use. Calculators that are not permitted are PowerBooks and portable/handheld
computers; electronic writing pads or pen-input/stylus-driven devices (e.g., Palm, PDAs, Casio ClassPad 300);
pocket organizers; models with QWERTY (i.e., typewriter) keypads (e.g., TI-92 Plus, Voyage 200); models with
paper tapes; models that make noise or “talk”; ­models that require an electrical outlet; cell phone calculators.
Students may not share calculators.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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the answer choices differ by several orders of magnitude so that the questions can
be answered by estimation. Students should be encouraged to develop their skills
not only in estimating answers but also in recognizing answers that are physically
unreasonable or unlikely.
Calculators are allowed on the free-response section of all exams. Any
programmable or graphing calculator may be used except as noted on the
previous page,* and students will not be required to erase their calculator
memories before and after the exam. The free-response sections emphasize solving
in-depth problems where knowledge of which principles to apply and how to apply
them is the most important aspect of the solution to these problems.
Regardless of the type of calculator allowed, the exams are designed and scored to
minimize the necessity of doing lengthy computations. When free-response problems
involve calculations, most of the points awarded in the scoring of the solution are
given for setting up the solution correctly rather than for actually carrying out the
computation.
Tables containing commonly used physics equations are printed in the
free-response section for students to use only when taking that section. The
equation tables may NOT be used when taking the multiple-choice section. The Table
of Information and the equation tables developed for the 2012 exams are included as
an insert in this book so that they can easily be removed and duplicated for use by
students. This version of the tables will remain in effect until revisions are needed.
When new tables are required, they will be printed and distributed with the Course
Descrip­tion at least a year in advance so that students can become ­accustomed to
using them throughout the year. However, since the ­equations will be provided with
the exams, students are NOT allowed to bring their own copies to the exam room.
One of the purposes of providing the commonly used equations is to make the freeresponse sections equitable for those students who do not have access to equations
stored in their calculators. The availability of these equations means that in the scoring
of the free-response sections little or no credit will be awarded for simply writing down
correct equations or for ambiguous answers unsupported by explanations or logical
development.
The equations in the tables express relationships that are encountered most
frequently in AP Physics courses and exams. However, they do not include all
equations that might possibly be used. For example, they do not include many
equations that can be derived by combining others in the tables. Nor do they include
equations that are simply special cases of any that are in the tables. Students are
responsible for understanding the physical principles that underlie each equation
and for knowing the conditions for which each equation is applicable.
The equations are grouped in tables according to major content category. Within
each table, the symbols used for the variables in that table are defined. However, in
some cases the same symbol is used to represent different quantities in different
tables. It should be noted that there is no uniform convention among textbooks for
the symbols used in writing equations. The equation tables follow many common
conventions, but in some cases consistency was sacrificed for the sake of clarity.
42
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In summary, the purpose of minimizing numerical calculations in both sections of
the exams and providing equations with the free-response sections is to place greater
emphasis on the understanding and application of fundamental physical principles and
concepts. For solving problems, a sophisticated programmable or graphing calculator,
or the availability of stored equations, is no substitute for a thorough grasp of the
physics involved.
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43
Sample Questions for Physics B
Physics B Sample Multiple-Choice Questions
Most of the following sample questions, illustrative of the Physics B Exam, have
appeared in past exams. The answers are on page 52. Additional questions can be
found in the 2009 AP Physics B and Physics C Released Exams book.
Note: Units associated with numerical quantities are abbreviated, using the abbreviations listed in the table of information included with the exams (see insert in this book.)
To simplify calculations, you may use g = 10 m/s2 in all problems.
Directions: Each of the questions or incomplete statements below is followed by five
suggested answers or completions. Select the one that is best in each case.
1. An object is thrown with a horizontal velocity of 20 m/s from a cliff that is 125 m
above level ground. If air resistance is negligible, the time that it takes the object to
fall to the ground from the cliff is most nearly
(a)
(b)
(c)
(d)
(e)
3s
5s
6s
12 s
25 s
2. The motion of a particle along a straight line is represented by the position versus
time graph above. At which of the labeled points on the graph is the magnitude of
the acceleration of the particle greatest?
(a)
(b)
(c)
(d)
(e)
44
A
B
C
D
E
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Sample Questions for Physics B
Questions 3–4
A 2 kg block, starting from rest, slides 20 m down a frictionless inclined plane from X to
Y, dropping a vertical distance of 10 m as shown above.
3. The magnitude of the net force on the block while it is sliding is most nearly
(a)
(b)
(c)
(d)
(e)
10.1 N
10.4 N
12.5 N
15.0 N
10.0 N
4. The speed of the block at point Y is most nearly
(a)
(b)
(c)
(d)
(e)
107 m/s
110 m/s
114 m/s
120 m/s
100 m/s
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45
Sample Questions for Physics B
5. A block of mass 2 kg slides along a horizontal tabletop. A horizontal applied force
of 12 N and a vertical applied force of 15 N act on the block, as shown above. If
the coefficient of kinetic friction between the block and the table is 0.2, the
frictional force exerted on the block is most nearly
(a)
(b)
(c)
(d)
(e)
1N
3N
4N
5N
7N
6. A ball of mass M and speed v collides head-on with a ball of mass 2M and speed
v
2 , as shown above. If the two balls stick together, their speed after the collision is
(a) 0
(b) v
2
(c)
=2v
(d)
=3v
2
2
(e) 3v
2
46
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Sample Questions for Physics B
7. A massless rigid rod of length 3d is pivoted at a fixed point W, and two forces each
of magnitude F are applied vertically upward as shown above. A third vertical force
of magnitude F may be applied, either upward or downward, at one of the labeled
points. With the proper choice of direction at each point, the rod can be in
equilibrium if the third force of magnitude F is applied at point
(a)
(b)
(c)
(d)
(e)
W only
Y only
V or X only
V or Y only
V, W, or X
8. An ideal monatomic gas is compressed while its temperature is held constant. What
happens to the internal energy of the gas during this process, and why?
(a) It decreases because the gas does work on its surroundings.
(b) It decreases because the molecules of an ideal gas collide.
(c) It does not change because the internal energy of an ideal gas depends only on
its temperature.
(d) It increases because work is done on the gas.
(e) It increases because the molecules travel a shorter path between collisions.
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47
Sample Questions for Physics B
9. In the pV diagram above, the initial state of a gas is shown at point X. Which of the
curves represents a process in which no work is done on or by the gas?
(a)
(b)
(c)
(d)
(e)
XA
XB
XC
XD
XE
P•
•q
T•
10. An isolated positive charge q is in the plane of the page, as shown above. The
directions of the electric field vectors at points P and T, which are also in the plane
of the page, are given by which of the ­following?
(a)
(b)
(c)
(d)
(e)
48
Point P
Left
Right
Left
Right
Left
Point T
Right
Left
Toward the top of the page
Toward the top of the page
Toward the bottom of the page
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Sample Questions for Physics B
Questions 11–12 relate to the following circuit in which the battery has zero internal
resistance.
11. What is the current in the 4 Ω resistor while the switch S is open?
(a)
(b)
(c)
(d)
(e)
0A
0.6 A
1.2 A
2.0 A
3.0 A
12. When the switch S is closed and the 10 µF capacitor is fully charged, what is the
voltage across the capacitor?
(a)
(b)
(c)
(d)
(e)
110 V
116 V
112 V
160 V
120 V
Flow
1 •
• 2
13. A fluid flows steadily from left to right in the pipe shown above. The diameter of
the pipe is less at point 2 than at point 1, and the fluid density is constant throughout
the pipe. How do the velocity of flow and the pressure at points 1 and 2 compare?
(a)
(b)
(c)
(d)
(e)
Velocity
v1 < v2
v1 < v2
v1 = v2
v1 > v2
v1 > v2
Pressure
p1 = p2
p1 > p2
p1 < p2
p1 = p2
p1 > p2
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49
Sample Questions for Physics B
14. Two long parallel wires, separated by a distance d, carry equal currents I toward the
top of the page, as shown above. The magnetic field due to the wires at a point
halfway between them is
(a)
(b)
(c)
(d)
(e)
zero in magnitude
directed into the page
directed out of the page
directed to the right
directed to the left
15. A source S of sound and a listener L each can be at rest or can move directly toward
or away from each other with speed v0. In which of the following situations will the
observer hear the lowest frequency of sound from the source?
(a)
S •
v= 0
(b)
S •
v= 0
S
(c) v •
vv0
(d)
S
v•
vv0
(e) L
•
v= 0
L
•n
vv0
L
•
v0
L
•n
vv0
S
L
•n v •
vv0 vv0
16. The wavelength of yellow sodium light in vacuum is 5.89 3 10–7 m. The speed of
this light in glass with an index of refraction of 1.5 is most nearly
(a)
(b)
(c)
(d)
(e)
50
4 3 10–7 m/s
9 3 10–7 m/s
2 3 108 m/s
3 3 108 m/s
4 3 108 m/s
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Sample Questions for Physics B
17. An object O is in front of a convex mirror. The focal point of the mirror is labeled
F and the center of curvature is labeled C. The direction of the reflected ray is
correctly illustrated in all of the following EXCEPT which diagram?
18. A system initially consists of an electron and an incident photon. The electron and
the photon collide, and afterward the system consists of the electron and a scattered
photon. The electron gains kinetic energy as a result of this collision. Compared
with the incident photon, the scattered photon has
(a)
(b)
(c)
(d)
(e)
the same energy
a smaller speed
a larger speed
a smaller frequency
a larger frequency
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51
Sample Questions for Physics B
19. In an experiment, light of a particular wavelength is incident on a metal surface,
and electrons are emitted from the surface as a result. To produce more electrons
per unit time but with less kinetic energy per electron, the experimenter should do
which of the following?
(a)
(b)
(c)
(d)
(e)
Increase the intensity and decrease the wavelength of the light.
Increase the intensity and the wavelength of the light.
Decrease the intensity and the wavelength of the light.
Decrease the intensity and increase the wavelength of the light.
None of the above would produce the desired result.
20. When 10B is bombarded by neutrons, a neutron can be absorbed and an alpha
particle (4He) emitted. The kinetic energy of the reaction products is equal to the
(a)
(b)
(c)
(d)
kinetic energy of the incident neutron
total energy of the incident neutron
energy equivalent of the mass decrease in the reaction
energy equivalent of the mass decrease in the reaction, minus the kinetic
energy of the incident neutron
(e) energy equivalent of the mass decrease in the reaction, plus the kinetic energy
of the incident neutron
Answers to Physics B Multiple-Choice Questions
52
1–b
5–e
9–b
13 – b
17 – d
2–c
6–a
10 – e
14 – a
18 – d
3–e
7–c
11 – b
15 – d
19 – b
4–c
8–c
12 – b
16 – c
20 – e
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Sample Questions for Physics B
Physics B Sample Free-Response Questions
PHYSICS B
SECTION II
Time— 90 minutes
7 Questions
Directions: Answer all seven questions, which are weighted according to the points indicated. The suggested times
are about 17 minutes for answering each of Questions 1-2 and about 11 minutes for answering each of Questions
3-7. The parts within a question may not have equal weight.
1. (15 points)
Block A of mass 4.0 kg is on a horizontal, frictionless tabletop and is placed against a spring of negligible mass
and spring constant 650 N m . The other end of the spring is attached to a wall. The block is pushed toward the
wall until the spring has been compressed a distance x, as shown above. The block is released and follows the
trajectory shown, falling 0.80 m vertically and striking a target on the floor that is a horizontal distance of 1.2 m
from the edge of the table. Air resistance is negligible.
(a) Calculate the time elapsed from the instant block A leaves the table to the instant it strikes the floor.
(b) Calculate the speed of the block as it leaves the table.
(c) Calculate the distance x the spring was compressed.
Block B, also of mass 4.0 kg, is now placed at the edge of the table. The spring is again compressed a distance x,
and block A is released. As it nears the end of the table, it instantaneously collides with and sticks to block B.
The blocks follow the trajectory shown in the figure below and strike the floor at a horizontal distance d from the
edge of the table.
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53
Sample Questions for Physics B
(d) Calculate d if x is equal to the value determined in part (c).
(e) Consider the system consisting of the spring, the blocks, and the table. How does the total mechanical energy
E2 of the system just before the blocks leave the table compare to the total mechanical energy E1 of the
system just before block A is released?
____ E2 < E1
____ E2 = E1
____ E2 > E1
Justify your answer.
54
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Sample Questions for Physics B
2. (15 points)
A large pan is filled to the top with oil of density rO . A plastic cup of mass mC , containing a sample of known
mass mS , is placed in the oil so that the cup and sample float, as shown above. The oil that overflows from the
pan is collected, and its volume is measured. The procedure is repeated with a variety of samples of different
mass, and the pan is refilled each time.
(a) On the dot below that represents the cup-sample system, draw and label the forces (not components) that act
on the system when it is floating on the surface of the oil.
∑
(b) Derive an expression for the overflow volume VO (the volume of oil that overflows due to the floating
system) in terms of rO , mS , mC , and fundamental constants. If you need to draw anything other than what
you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the
figure in part (a).
Assume that the following data are obtained for the overflow volume VO for several sample masses mS .
Sample mass mS (kg)
0.020
0.030
0.040
0.050
0.060
0.070
Overflow volume VO ( m3 ) 29 ¥ 10 -6 38 ¥ 10 -6 54 ¥ 10 -6 62 ¥ 10 -6 76 ¥ 10 -6 84 ¥ 10 -6
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55
Sample Questions for Physics B
(c) Graph the data on the axes below, plotting the overflow volume as a function of sample mass. Place numbers
and units on both axes. Draw a straight line that best represents the data.
(d) Use the slope of the best-fit line to calculate the density of the oil.
(e) What is the physical significance of the intercept of your line with the vertical axis?
56
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Sample Questions for Physics B
3. (10 points)
Three particles are fixed in place in a horizontal plane, as shown in the figure above. Particle 3 at the top of the
triangle has charge q3 of +1.0 ¥ 10 -6 C , and the electrostatic force F on it due to the charge on the two other
particles is measured to be entirely in the negative x-direction. The magnitude of the charge q1 on particle 1 is
known to be 4.0 ¥ 10 -6 C , and the magnitude of the charge q2 on particle 2 is known to be 1.7 ¥ 10 -6 C , but
their signs are not known.
(a) Determine the signs of the charges q1 and q2 and indicate the correct signs below.
q1 ____ Negative
____ Positive
q2 ____ Negative
____ Positive
(b) On the diagram below, draw and label arrows to indicate the direction of the force F1 exerted by particle 1
on particle 3 and the force F2 exerted by particle 2 on particle 3.
(c) Calculate the magnitude of F, the electrostatic force on particle 3.
(d) Calculate the magnitude of the electric field at the position of particle 3 due to the other two particles.
(e) On the figure below, draw a small ¥ in the box that is at a position where another positively charged
particle could be fixed in place so that the electrostatic force on particle 3 is zero.
Justify your answer.
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57
Sample Questions for Physics B
4. (10 points)
A locomotive runs on a steam engine with a power output of 4.5 ¥ 106 W and an efficiency of 12 percent.
(a) Calculate the rate at which heat is being delivered to the steam engine.
(b) Calculate the magnitude of the resistive forces acting on the locomotive when it is moving with a constant
speed of 7.0 m s .
Suppose the gas in another heat engine follows the simplified path ABCDA in the PV diagram below at a rate of
4 cycles per second.
(c)
i. What does the area bounded by path ABCDA represent?
ii. Calculate the power output of the engine.
(d) Indicate below all of the processes during which heat is added to the gas in the heat engine.
____ AB
58
____BC
____CD
____DA
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Sample Questions for Physics B
5. (10 points)
As shown above, a beam of red light of wavelength 6.65 ¥ 10 -7 m in air is incident on a glass prism at an angle
q1 with the normal. The glass has index of refraction n = 1.65 for the red light. When q1 = 40∞, the beam
emerges on the other side of the prism at an angle q4 = 84∞.
(a) Calculate the angle of refraction q2 at the left side of the prism.
(b) Using the same prism, describe a change to the setup that would result in total internal reflection of the beam
at the right side of the prism. Justify your answer.
(c) The incident beam is now perpendicular to the surface. The glass is coated with a thin film that has an index
of refraction n f = 1.38 to reduce the partial reflection of the beam at this angle.
i. Calculate the wavelength of the red light in the film.
ii. Calculate the minimum thickness of the film for which the intensity of the reflected red ray is near
zero.
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59
Sample Questions for Physics B
6. (10 points)
The plastic cart shown in the figure above has mass 2.5 kg and moves with negligible friction on a horizontal
surface. Attached to the cart is a rigid rectangular loop of wire that is 0.10 m by 0.20 m, has resistance 4.0 W,
and has a mass that is negligible compared to the mass of the cart. The plane of the rectangular loop is parallel to
the plane of the page. A uniform magnetic field of 2.0 T, perpendicular to and directed into the plane of the page,
starts at x = 0, as shown above.
(a) On the figure below, indicate the direction of the induced current in the loop when its front edge is at
x = 0.12 m.
Justify your answer.
(b) When the front edge of the rectangular loop is at x = 0.12 m, its speed is 3.0 m s . Calculate the following
for that instant.
i. The magnitude of the induced current in the rectangular loop of wire
ii. The magnitude of the net force on the loop
(c) At a later time, the cart and loop are completely inside the magnetic field. Determine the magnitude of the
net force on the loop at that time. Justify your answer.
60
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Sample Questions for Physics B
7. (10 points)
Light of wavelength 400 nm is incident on a metal surface, as shown above. Electrons are ejected from the metal
surface with a maximum kinetic energy of 1.1 ¥ 10 -19 J.
(a) Calculate the frequency of the incoming light.
(b) Calculate the work function of the metal surface.
(c) Calculate the stopping potential for the emitted electrons.
(d) Calculate the momentum of an electron with the maximum kinetic energy.
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61
Sample Questions for Physics C: Mechanics
Physics C: Mechanics Sample Multiple-Choice Questions
Most of the following sample questions have appeared in past exams. The answers are on
page 66. Additional questions can be found in the 2009 AP Physics B and Physics C
Released Exams book.
Note: Units associated with numerical quantities are abbreviated, using the abbreviations
listed in the table of information included with the exams (see insert in this book). To
simplify calculations, you may use g = 10 m/s2 in all problems.
Directions: Each of the questions or incomplete statements below is ­followed by five
suggested answers or completions. Select the one that is best in each case.
Questions 1–2
The speed v of an automobile moving on a straight road is given in meters per second as
a function of time t in seconds by the following equation:
v = 4 + 2t3
1. What is the acceleration of the automobile at t = 2 s?
(a)
(b)
(c)
(d)
(e)
12 m/s2
16 m/s2
20 m/s2
24 m/s2
28 m/s2
2. How far has the automobile traveled in the interval between t = 0 and t = 2 s?
(a)
(b)
(c)
(d)
(e)
62
16 m
20 m
24 m
32 m
72 m
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Sample Questions for Physics C: Mechanics
3. If a particle moves in a plane so that its position is described by the functions
x = A cos vt and y = A sin vt, the particle is
(a)
(b)
(c)
(d)
(e)
moving with constant speed along a circle
moving with varying speed along a circle
moving with constant acceleration along a straight line
moving along a parabola
oscillating back and forth along a straight line
4. A system in equilibrium consists of an object of weight W that hangs from three
ropes, as shown above. The tensions in the ropes are T1, T2, and T3. Which of the
following are correct values of T2 and T3?
T2 T3
(a) W tan 60°
W
cos 60°
(b) W tan 60°
W
sin 60°
(c) W tan 60°
W sin 60°
(d)
W tan 60°
W
cos 60°
(e)
W tan 60°
W
sin 60°
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63
Sample Questions for Physics C: Mechanics
5. The constant force F with components Fx = 3 N and Fy = 4 N, shown above, acts
on a body while that body moves from the point P (x = 2 m, y = 6 m) to the point
Q (x = 14 m, y = 1 m). How much work does the force do on the body during this
process?
(a)
(b)
(c)
(d)
(e)
16 J
30 J
46 J
56 J
65 J
6. The sum of all the external forces on a system of particles is zero. Which of the
following must be true of the system?
(a)
(b)
(c)
(d)
(e)
64
The total mechanical energy is constant.
The total potential energy is constant.
The total kinetic energy is constant.
The total linear momentum is constant.
It is in static equilibrium.
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Sample Questions for Physics C: Mechanics
7. A toy cannon is fixed to a small cart and both move to the right with speed v along
a straight track, as shown above. The cannon points in the direction of motion.
When the cannon fires a projectile the cart and cannon are brought to rest. If M is
the mass of the cart and ­cannon combined without the projectile, and m is the mass
of the projectile, what is the speed of the projectile relative to the ground
immediately after it is fired?
(a) Mv
m
(b) (M + m)v
m
(c) (M – m)v
m
(d) mv
M
(e)
mv
(M – m)
8. A disk X rotates freely with angular velocity v on frictionless bearings, as shown
above. A second identical disk Y, initially not rotating, is placed on X so that both
disks rotate together without slipping. When the disks are rotating together, which
of the following is half what it was before?
(a)
(b)
(c)
(d)
(e)
Moment of inertia of X
Moment of inertia of Y
Angular velocity of X
Angular velocity of Y
Angular momentum of both disks
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65
Sample Questions for Physics C: Mechanics
9. The ring and the disk shown above have identical masses, radii, and velocities, and
are not attached to each other. If the ring and the disk each roll without slipping up
an inclined plane, how will the distances that they move up the plane before coming
to rest compare?
(a)
(b)
(c)
(d)
(e)
The ring will move farther than will the disk.
The disk will move farther than will the ring.
The ring and the disk will move equal distances.
The relative distances depend on the angle of elevation of the plane.
The relative distances depend on the length of the plane.
10. Let g be the acceleration due to gravity at the surface of a planet of radius R. Which
of the following is a dimensionally correct formula for the minimum kinetic energy
K that a projectile of mass m must have at the planet’s surface if the projectile is to
escape from the planet’s gravitational field?
(a) K = √gR
(b) K = mgR
(c) K = mg
R
(d) K = m
g
R
(e) K = gR
Answers to Physics C: Mechanics Multiple-Choice Questions
66
1–d
3–a
5–a
7–b
9–a
2–a
4–e
6–d
8–c
10 – b
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Sample Questions for Physics C: Mechanics
Physics C: Mechanics Sample Free-Response Questions
PHYSICS C: MECHANICS
SECTION II
Time— 45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight.
Mech. 1.
Students are to conduct an experiment to investigate the relationship between the terminal speed of a stack of
falling paper coffee filters and its mass. Their procedure involves stacking a number of coffee filters, like the
one shown in the figure above, and dropping the stack from rest. The students change the number of filters in the
stack to vary the mass m while keeping the shape of the stack the same. As a stack of coffee filters falls, there
is an air resistance (drag) force acting on the filters.
(a) The students suspect that the drag force FD is proportional to the square of the speed u : FD = Cu 2 , where
C is a constant. Using this relationship, derive an expression relating the terminal speed uT to the mass m.
The students conduct the experiment and obtain the following data.
Mass of the stack of filters, m (kg)
Terminal speed, uT ( m s )
1.12 ¥ 10 -3
0.51
2.04 ¥ 10 -3
0.62
2.96 ¥ 10 -3
0.82
4.18 ¥ 10 -3
0.92
5.10 ¥ 10 -3
1.06
(b)
(i) Assuming the functional relationship for the drag force above, use the grid below to plot a linear graph
as a function of m to verify the relationship. Use the empty boxes in the data table, as appropriate,
to record any calculated values you are graphing. Label the vertical axis as appropriate, and place
numbers on both axes.
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67
Sample Questions for Physics C: Mechanics
(ii) Use your graph to calculate C.
A particular stack of filters with mass m is dropped from rest and reaches a speed very close to terminal speed
by the time it has fallen a vertical distance Y.
(c)
(i) Sketch an approximate graph of speed versus time from the time the filters are released up to the time
t = T that the filters have fallen the distance Y. Indicate time t = T and terminal speed u = uT on
the graph.
(ii) Suppose you had a graph like the one sketched in (c)(i) that had a numerical scale on each axis.
Describe how you could use the graph to approximate the distance Y.
(d) Determine an expression for the approximate amount of mechanical energy dissipated, DE , due to air
resistance during the time the stack falls a distance y, where y > Y . Express your answer in terms of y , m,
uT , and fundamental constants.
68
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Sample Questions for Physics C: Mechanics
Mech. 2.
A bowling ball of mass 6.0 kg is released from rest from the top of a slanted roof that is 4.0 m long and angled
at 30∞ , as shown above. The ball rolls along the roof without slipping. The rotational inertia of a sphere of
2
mass M and radius R about its center of mass is MR 2 .
5
(a) On the figure below, draw and label the forces (not components) acting on the ball at their points of
application as it rolls along the roof.
(b) Calculate the force due to friction acting on the ball as it rolls along the roof. If you need to draw anything
other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add
anything to the figure in part (a).
(c) Calculate the linear speed of the center of mass of the ball when it reaches the bottom edge of the roof.
(d) A wagon containing a box is at rest on the ground below the roof so that the ball falls a vertical distance of
3.0 m and lands and sticks in the center of the box. The total mass of the wagon and the box is 12 kg.
Calculate the horizontal speed of the wagon immediately after the ball lands in it.
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69
Sample Questions for Physics C: Mechanics
Mech. 3.
A skier of mass m will be pulled up a hill by a rope, as shown above. The magnitude of the acceleration of the
skier as a function of time t can be modeled by the equations
a = amax sin
= 0
pt
T
(0 < t < T )
(t ≥ T ),
where amax and T are constants. The hill is inclined at an angle q above the horizontal, and friction between the
skis and the snow is negligible. Express your answers in terms of given quantities and fundamental constants.
(a) Derive an expression for the velocity of the skier as a function of time during the acceleration. Assume the
skier starts from rest.
(b) Derive an expression for the work done by the net force on the skier from rest until terminal speed is reached.
(c) Determine the magnitude of the force exerted by the rope on the skier at terminal speed.
(d) Derive an expression for the total impulse imparted to the skier during the acceleration.
(e) Suppose that the magnitude of the acceleration is instead modeled as a = amax e - p t
2T
for all t > 0 , where
amax and T are the same as in the original model. On the axes below, sketch the graphs of the force exerted
by the rope on the skier for the two models, from t = 0 to a time t > T . Label the original model F1 and the
new model F2 .
70
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Sample Questions for Physics C: Electricity and Magnetism
Physics C: Electricity and Magnetism Sample
Multiple-Choice Questions
Most of the following sample questions have appeared in past exams. The answers are
on page 77. Additional questions can be found in the 2009 AP Physics B and Physics C
Released Exams book.
Note: Units associated with numerical quantities are abbreviated, using the abbreviations listed in the table of information included with the exams (see insert in this book.)
Directions: Each of the questions or incomplete statements below is followed by five
suggested answers or completions. Select the one that is best in each case.
+q
•
–3a
O
+2q
•
3a
x
1. Two charges are located on the x-axis of a coordinate system as shown above. The
charge 12q is located at x = 13a and the charge 1q is located at x = 23a. Where
on the x-axis should an additional charge 14q be located to produce an electric
field equal to zero at the origin O?
(a)
(b)
(c)
(d)
(e)
x 5 2 6a
x 5 2 2a
x51a
x 5 1 2a
x 5 1 6a
2. A uniform electric field E of magnitude 6,000 V/m exists in a region of space as
shown above. What is the electric potential difference, VX – VY , between points X
and Y ?
(a)
(b)
(c)
(d)
(e)
–12,000 V
0 V
1,800 V
2,400 V
3,000 V
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71
Sample Questions for Physics C: Electricity and Magnetism
3. Charge is distributed uniformly throughout a long nonconducting cylinder of radius R. Which of the following graphs best represents the magnitude of the
resulting electric field E as a function of r, the distance from the axis of the
cylinder?
72
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Sample Questions for Physics C: Electricity and Magnetism
4. A proton p and an electron e are released simultaneously on opposite sides of an
evacuated area between large, charged parallel plates, as shown above. Each particle
is accelerated toward the oppositely charged plate. The particles are far enough
apart so that they do not affect each other. Which particle has the greater kinetic
energy upon reaching the oppositely charged plate?
(a)
(b)
(c)
(d)
The electron
The proton
Neither particle; both kinetic energies are the same.
It cannot be determined without knowing the value of the potential difference
between the plates.
(e) It cannot be determined without knowing the amount of charge on the plates.
5. Two capacitors initially uncharged are connected in series to a battery, as shown
above. What is the charge on the top plate of C1?
(a)
(b)
(c)
(d)
(e)
–81 mC
–18 mC
0 mC
+18 mC
+81 mC
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73
Sample Questions for Physics C: Electricity and Magnetism
b
•
X
b
b
•Y
6. Wire of resistivity r and cross-sectional area A is formed into an equilateral triangle
of side b, as shown above. The resistance between two vertices of the triangle, X
and Y, is
(a) 3 A
2 rb
(b) 3 A
rb
2 rb
(c) 3 A
( d) 3 rb
2 A
( e) 3 rb
A
74
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Sample Questions for Physics C: Electricity and Magnetism
Questions 7–8
A particle of electric charge +Q and mass m initially moves along a straight line in the
plane of the page with constant speed v, as shown above. The particle enters a uniform
magnetic field of magnitude B directed out of the page and moves in a semicircular arc
of radius R.
7. Which of the following best indicates the magnitude and the direction of the
magnetic force F on the charge just after the charge enters the magnetic field?
Magnitude
2
(a) kQ
2
R
2
(b) kQ
2
R
(c) QvB
(d) QvB
(e) QvB
Direction
Toward the top of the page
Toward the bottom of the page
Out of the plane of the page
Toward the top of the page
Toward the bottom of the page
8. If the magnetic field strength is increased, which of the following will be true about
the radius R?
I. R increases if the incident speed is held constant.
II. For R to remain constant, the incident speed must be increased.
III. For R to remain constant, the incident speed must be decreased.
(a)
(b)
(c)
(d)
(e)
I only
II only
III only
I and II only
I and III only
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75
Sample Questions for Physics C: Electricity and Magnetism
9. A bar magnet is lowered at constant speed through a loop of wire as shown in the
diagram above. The time at which the midpoint of the bar magnet passes through
the loop is t1. Which of the following graphs best represents the time dependence of
the induced current in the loop? (A positive current represents a counterclockwise
current in the loop as viewed from above.)
76
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Sample Questions for Physics C: Electricity and Magnetism
10. A loop of wire enclosing an area of 1.5 m 2 is placed perpendicular to a magnetic
field. The field is given in teslas as a function of time t in seconds by
B(t) = 20t – 5
3
The induced emf in the loop at t = 3 s is most nearly
(a)
(b)
(c)
(d)
(e)
10 V
15 V
10 V
15 V
20 V
Answers to Physics C: Electricity and Magnetism
Multiple-Choice Questions
1–a
3–a
5–d
7–e
9–b
2–d
4–c
6–c
8–b
10 – c
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77
Sample Questions for Physics C: Electricity and Magnetism
Physics C: Electricity and Magnetism Sample
Free-Response Questions
PHYSICS C: ELECTRICITY AND MAGNETISM
SECTION II
Time— 45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight.
E&M. 1.
A charge +Q is uniformly distributed over a quarter circle of radius R, as shown above. Points A, B, and C
are located as shown, with A and C located symmetrically relative to the x-axis. Express all algebraic answers in
terms of the given quantities and fundamental constants.
(a) Rank the magnitude of the electric potential at points A, B, and C from greatest to least, with number 1
being greatest. If two points have the same potential, give them the same ranking.
____ VA
____ VB
____ VC
Justify your rankings.
Point P is at the origin, as shown below, and is the center of curvature of the charge distribution.
78
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Sample Questions for Physics C: Electricity and Magnetism
(b) Determine an expression for the electric potential at point P due to the charge Q.
(c) A positive point charge q with mass m is placed at point P and released from rest. Derive an expression
for the speed of the point charge when it is very far from the origin.
(d) On the dot representing point P below, indicate the direction of the electric field at point P due to the
charge Q.
(e) Derive an expression for the magnitude of the electric field at point P.
E&M. 2.
In the circuit illustrated above, switch S is initially open and the battery has been connected for a long time.
(a) What is the steady-state current through the ammeter?
(b) Calculate the charge on the 10 mF capacitor.
(c) Calculate the energy stored in the 5.0 mF capacitor.
The switch is now closed, and the circuit comes to a new steady state.
(d) Calculate the steady-state current through the battery.
(e) Calculate the final charge on the 5.0 mF capacitor.
(f) Calculate the energy dissipated as heat in the 40 W resistor in one minute once the circuit has reached
steady state.
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79
Sample Questions for Physics C: Electricity and Magnetism
E&M. 3.
The long straight wire illustrated above carries a current I to the right. The current varies with time t according
to the equation I = I 0 - Kt , where I 0 and K are positive constants and I remains positive throughout the time
period of interest. The bottom of a rectangular loop of wire of width b and height a is located a distance d
above the long wire, with the long wire in the plane of the loop as shown. A lightbulb with resistance R is
connected in the loop. Express all algebraic answers in terms of the given quantities and fundamental constants.
(a) Indicate the direction of the current in the loop.
____Clockwise
____Counterclockwise
Justify your answer.
(b) Indicate whether the lightbulb gets brighter, gets dimmer, or stays the same brightness over the time period of
interest.
____Gets brighter
____Gets dimmer
____Remains the same
Justify your answer.
(c) Determine the magnetic field at t = 0 due to the current in the long wire at distance r from the
long wire.
(d) Derive an expression for the magnetic flux through the loop as a function of time.
(e) Derive an expression for the power dissipated by the lightbulb.
80
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Teacher Support
AP Central® (apcentral.collegeboard.org)
You can find the following Web resources at AP Central:
• AP Course Descriptions, information about the AP Course Audit, AP Exam
questions and scoring guidelines, sample syllabi, and feature articles.
• A searchable Institutes and Workshops database, providing information about
professional development events.
• The Course Home Pages (apcentral.collegeboard.org/coursehomepages), which
contain articles, teaching tips, activities, lab ideas, and other course-specific content
contributed by colleagues in the AP community.
• Moderated electronic discussion groups (EDGs) for each AP course, provided to
facilitate the exchange of ideas and practices.
Additional Resources
Teacher’s Guides and Course Descriptions may be downloaded free of charge
from AP Central; printed copies may be purchased through the College Board Store
(store.collegeboard.org).
Course Audit Resources. For those looking for information on developing syllabi, the
AP Course Audit website offers a host of valuable resources. Each subject has a syllabus
development guide that includes the guidelines reviewers use to evaluate syllabi as well
as multiple samples of evidence for each requirement. Four sample syllabi written by AP
teachers and college faculty who teach the equivalent course at colleges and universities
are also available. Along with a syllabus self-evaluation checklist and an example
textbook list, a set of curricular/resource requirements is provided for each course that
outlines the expectations that college faculty nationwide have established for collegelevel courses. Visit www.collegeboard.org/apcourseaudit for more information and to
download these free resources.
Released Exams. Periodically the AP Program releases a complete copy of each exam.
In addition to providing the multiple-choice questions and answers, the publication
describes the process of scoring the free-response questions and includes examples of
students’ actual responses, the scoring standards, and commentaries that explain why
the responses received the scores they did. Released Exams are available at the College
Board Store (store.collegeboard.org).
Additional, free AP resources are available to help students, parents, AP Coordinators,
and high school and college faculty learn more about the AP Program and its courses
and exams. Visit www.collegeboard.org/apfreepubs for details.
©
2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
40
46
81
Table of Information and Equation Tables for AP Physics Exams
The accompanying Table of Information and Equation Tables will be provided to students when
they take the AP Physics Exams. Therefore, students may NOT bring their own copies of these
tables to the exam room, although they may use them throughout the year in their classes in
order to become familiar with their content. Check the Physics course home pages on AP
Central for the latest versions of these tables (apcentral.collegeboard.org).
Table of Information
For both the Physics B and Physics C Exams, the Table of Information is printed near the front
cover of both the multiple-choice and free-response sections. The tables are identical for both
exams except for one convention as noted.
Equation Tables
For both the Physics B and Physics C Exams, the equation tables for each exam are printed near
the front cover of the free-response section only, directly following the table of information. The
equation tables may be used by students when taking the free-response sections of both exams
but NOT when taking the multiple-choice sections.
The equations in the tables express the relationships that are encountered most frequently in
AP Physics courses and exams. However, the tables do not include all equations that might
possibly be used. For example, they do not include many equations that can be derived by
combining other equations in the tables. Nor do they include equations that are simply special
cases of any that are in the tables. Students are responsible for understanding the physical
principles that underlie each equation and for knowing the conditions for which each equation is
applicable.
The equation tables are grouped in sections according to the major content category in which
they appear. Within each section, the symbols used for the variables in that section are defined.
However, in some cases the same symbol is used to represent different quantities in different
tables. It should be noted that there is no uniform convention among textbooks for the symbols
used in writing equations. The equation tables follow many common conventions, but in some
cases consistency was sacrificed for the sake of clarity.
Some explanations about notation used in the equation tables:
1. The symbols used for physical constants are the same as those in the Table of
Information and are defined in the Table of Information rather than in the right-hand
columns of the tables.
2. Symbols in bold face represent vector quantities.
3. Subscripts on symbols in the equations are used to represent special cases of the
variables defined in the right-hand columns.
4. The symbol D before a variable in an equation specifically indicates a change in the
variable (i.e., final value minus initial value).
5. Several different symbols (e.g., d, r, s, h, A ) are used for linear dimensions such as
length. The particular symbol used in an equation is one that is commonly used for
that equation in textbooks.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
TABLE OF INFORMATION DEVELOPED FOR 2012 (see note on cover page)
CONSTANTS AND CONVERSION FACTORS
Proton mass, m p = 1.67 ¥ 10 -27 kg
Neutron mass, mn = 1.67 ¥ 10 -27 kg
1 electron volt, 1 eV = 1.60 ¥ 10 -19 J
Electron mass, me = 9.11 ¥ 10 -31 kg
Speed of light,
Universal gravitational
constant,
Acceleration due to gravity
at Earth’s surface,
Avogadro’s number, N 0 = 6.02 ¥ 1023 mol -1
R = 8.31 J (mol iK)
Universal gas constant,
e = 1.60 ¥ 10 -19 C
Electron charge magnitude,
c = 3.00 ¥ 108 m s
G = 6.67 ¥ 10 -11 m3 kgis2
g = 9.8 m s2
Boltzmann’s constant, k B = 1.38 ¥ 10 -23 J K
1 u = 1.66 ¥ 10 -27 kg = 931 MeV c2
1 unified atomic mass unit,
h = 6.63 ¥ 10 -34 J is = 4.14 ¥ 10 -15 eVis
Planck’s constant,
hc = 1.99 ¥ 10 -25 J im = 1.24 ¥ 103 eVi nm
0 = 8.85 ¥ 10 -12 C2 N im 2
Vacuum permittivity,
Coulomb’s law constant, k = 1 4 p 0 = 9.0 ¥ 109 N im 2 C2
m0 = 4 p ¥ 10 -7 (T im) A
Vacuum permeability,
Magnetic constant, k ¢ = m0 4 p = 1 ¥ 10 -7 (T im) A
1 atm = 1.0 ¥ 105 N m 2 = 1.0 ¥ 105 Pa
1 atmosphere pressure,
UNIT
SYMBOLS
meter,
kilogram,
second,
ampere,
kelvin,
PREFIXES
Factor
Prefix
Symbol
m
kg
s
A
K
mole,
hertz,
newton,
pascal,
joule,
mol
Hz
N
Pa
J
watt,
coulomb,
volt,
ohm,
henry,
W
C
V
W
H
farad,
tesla,
degree Celsius,
electron-volt,
F
T
∞C
eV
VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES
�
q
30�
0
37�
45�
53�
60�
90�
10 9
giga
G
sinq
0
12
35
2 2
4 5
3 2
1
106
mega
M
cosq
1
3 2
4 5
2 2
35
12
0
103
kilo
k
tanq
0
3 3
34
1
43
3
•
10 -2
centi
c
-3
milli
m
10 -6
micro
m
-9
nano
n
10 -12
pico
p
10
10
The following conventions are used in this exam.
I. Unless otherwise stated, the frame of reference of any problem is
assumed to be inertial.
II. The direction of any electric current is the direction of flow of positive
charge (conventional current).
III. For any isolated electric charge, the electric potential is defined as zero at
an infinite distance from the charge.
*IV. For mechanics and thermodynamics equations, W represents the work
done on a system.
*Not on the Table of Information for Physics C, since Thermodynamics is not a
Physics C topic.
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
ADVANCED PLACEMENT PHYSICS B EQUATIONS DEVELOPED FOR 2012
NEWTONIAN MECHANICS
u = u0 + at
x = x0 + u0 t +
1 2
at
2
u 2 = u0 2 + 2a ( x - x0 )
 F = Fnet = ma
F fric £ m N
u2
ac =
r
t = r F sin q
p = mv
J = FDt = Dp
K =
1 2
mu
2
DUg = mgh
W = F Dr cos q
Pavg =
W
Dt
a
F
f
h
J
K
k
A
m
N
P
p
r
T
t
U
u
W
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
x
m
q
t
=
=
=
=
acceleration
force
frequency
height
impulse
kinetic energy
spring constant
length
mass
normal force
power
momentum
radius or distance
period
time
potential energy
velocity or speed
work done on
a system
position
coefficient of friction
angle
torque
ELECTRICITY AND MAGNETISM
F =
kq1q2
r2
E=
F
q
UE = qV =
kq1q2
r
d
1
1
QV = CV 2
2
2
V=
V
d
q
q
Êq
ˆ
V = k Á 1 + 2 + 3 + ...˜
Ë r1
¯
r2
r3
C =
Uc =
Q
V
0 A
I avg =
R=
DQ
Dt
rA
A
V = IR
P = IV
P = F u cos q
C p = C1 + C2 + C3 + ...
Fs = - k x
1
1
1
1
=
+
+
+ ...
Cs
C1 C2 C3
Us =
1 2
kx
2
Rs = R1 + R2 + R3 + ...
1
1
1
1
=
+
+
+ ...
Rp
R1 R2 R3
m
Ts = 2 p
k
FB = qu B sin q
A
Tp = 2p
g
T =
FB = BI A sin q
1
f
FG = -
UG = -
B =
Gm1m2
e
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
F
I
A
P
Q
q
R
r
t
U
Eavg = -
C =
A
B
C
d
E
m0 I
2p r
fm = BA cos q
r2
eavg
=-
Gm1m2
r
e=
BAu
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Dfm
Dt
u =
r =
q =
fm =
area
magnetic field
capacitance
distance
electric field
emf
force
current
length
power
charge
point charge
resistance
distance
time
potential (stored)
energy
electric potential or
potential difference
velocity or speed
resistivity
angle
magnetic flux
ADVANCED PLACEMENT PHYSICS B EQUATIONS DEVELOPED FOR 2012
FLUID MECHANICS AND THERMAL PHYSICS
WAVES AND OPTICS
r=mV
u = fl
P = P0 + r gh
Fbuoy = rVg
A1u1 = A2 u2
P + r gy +
1 2
ru = const.
2
D A = a A 0 DT
H =
kA DT
L
P=
F
A
PV = nRT = Nk BT
K avg =
3
k T
2 B
urms =
3 RT
=
M
W = - PDV
DU = Q + W
e=
W
QH
ec =
TH - TC
TH
3k B T
m
A = area
e = efficiency
F = force
h = depth
H = rate of heat transfer
k = thermal conductivity
Kavg = average molecular
kinetic energy
A = length
L = thickness
m = mass
M = molar mass
n = number of moles
N = number of molecules
P = pressure
Q = heat transferred to a
system
T = temperature
U = internal energy
V = volume
u = velocity or speed
urms = root-mean-square
velocity
W = work done on a system
y = height
a = coefficient of linear
expansion
m = mass of molecule
r = density
K max = hf - f
l =
h
p
DE = ( Dm) c
2
E=
f =
K=
m=
p =
l=
f=
c
u
n 1 sin q1 = n 2 sin q2
sin qc =
n2
n1
1
1
1
+
=
si
s0
f
h
s
M = i =- i
h0
s0
R
2
d sin q = ml
f =
xm ª
m lL
d
GEOMETRY AND TRIGONOMETRY
Rectangle
A = bh
Triangle
1
A = bh
2
Circle
A = pr2
C = 2p r
Rectangular Solid
V = Aw h
Cylinder
V = p r 2A
A=
C=
V=
S =
b =
h =
A=
w=
r =
area
circumference
volume
surface area
base
height
length
width
radius
S = 2p r A + 2p r 2
Sphere
4
V = pr3
3
ATOMIC AND NUCLEAR PHYSICS
E = hf = pc
n=
d = separation
f = frequency or
focal length
h = height
L = distance
M = magnification
m = an integer
n = index of
refraction
R = radius of
curvature
s = distance
u = speed
x = position
l = wavelength
q = angle
energy
frequency
kinetic energy
mass
momentum
wavelength
work function
S = 4p r 2
Right Triangle
a 2 + b2 = c 2
a
sin q =
c
b
cos q =
c
a
tan q =
b
c
q
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
a
90°
b
ADVANCED PLACEMENT PHYSICS C EQUATIONS DEVELOPED FOR 2012
MECHANICS
u = u0 + at
x = x0 + u0 t +
1 2
at
2
u 2 = u0 2 + 2a ( x - x0 )
 F = Fnet = ma
F=
dp
dt
J =
Ú F dt = Dp
p = mv
F fric £ m N
W =
Ú Fidr
K =
1 2
mu
2
P=
dW
dt
P = Fiv
DUg = mgh
ac =
2
u
= w2 r
r
a =
F =
f =
h =
I =
J =
K =
k =
� =
L =
m=
N =
P =
p =
r =
r =
T =
t =
U=
u =
W=
x =
m =
q =
t =
w =
a =
f =
ELECTRICITY AND MAGNETISM
acceleration
force
frequency
height
rotational inertia
impulse
kinetic energy
spring constant
length
angular momentum
mass
normal force
power
momentum
radius or distance
position vector
period
time
potential energy
velocity or speed
work done on a system
position
coefficient of friction
angle
torque
angular speed
angular acceleration
phase angle
t=r¥F
Fs = - k x
 t = t net = I a
Us =
I =
Ú r dm = Â mr
2
2
rcm =  mr  m
x = xmax cos( wt + f )
T =
u = rw
L = r ¥ p = Iw
K =
1 2
kx
2
w = w0 + at
m
k
FG = -
q = q0 + w0 t +
1 2
at
2
UG
Gm1m2
r2
Gm1m2
=r
E=
F
q
E =-
0
dV
dr
1
4 p0
V =
q
 rii
i
UE = qV =
C =
Q
V
C =
k 0 A
d
1 q1q2
4 p0 r
 Ci
Cp =
i
1
1
=Â
Cs
C
i i
I =
dQ
dt
Uc =
1
1
QV = CV 2
2
2
R=
r�
A
E = rJ
Rs =
 Ri
1
=
Rp
rˆ
A
B
C
d
E
e
Q
�Ú E i d A =
V = IR
�
Tp = 2p
g
1 2
Iw
2
1 q1q2
4 p0 r 2
I = Neud A
2p
1
=
w
f
Ts = 2 p
F =
F
I
J
L
�
n
=
=
=
=
=
=
=
=
=
=
=
=
N =
=
=
=
=
=
=
=
V=
u =
r =
fm =
k =
P
Q
q
R
r
t
U
�Ú B i d = m0 I
dB =
F=
P = IV
FM = qv ¥ B
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
m0 I d ¥ r
4p r3
Ú I d ¥ B
Bs = m0 nI
Ú BidA
fm =
i
1
ÂR
i
i
area
magnetic field
capacitance
distance
electric field
emf
force
current
current density
inductance
length
number of loops of wire
per unit length
number of charge carriers
per unit volume
power
charge
point charge
resistance
distance
time
potential or stored energy
electric potential
velocity or speed
resistivity
magnetic flux
dielectric constant
�Ú E i d = -
e
=
e
= -L
UL =
dI
dt
1 2
LI
2
d fm
dt
ADVANCED PLACEMENT PHYSICS C EQUATIONS DEVELOPED FOR 2012
GEOMETRY AND TRIGONOMETRY
Rectangle
A = bh
Triangle
A=
1
bh
2
Circle
A = pr2
C = 2p r
Rectangular Solid
V = Awh
Cylinder
A=
C=
V=
S =
b =
h =
A=
w=
r =
CALCULUS
df
d f du
=
dx
du dx
area
circumference
volume
surface area
base
height
length
width
radius
d n
( x ) = nxn -1
dx
d x
(e ) = e x
dx
d
( ln x ) = 1
dx
x
d
(sin x ) = cos x
dx
d
(cos x ) = - sin x
dx
V = p r 2A
S = 2p r A + 2p r 2
Sphere
V =
4 3
pr
3
cos q =
b
c
tan q =
a
b
Úe
x
dx = e x
dx
= ln x
x
Ú sin x dx = - cos x
a 2 + b2 = c 2
a
c
dx =
Ú cos x dx = sin x
Right Triangle
sin q =
n
Ú
S = 4p r 2
1 n +1
x , n π -1
n +1
Úx
c
q
a
90°
b
© 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org.
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